MTH2008 REAL ANALYSIS Flashcards
FIELD PROPERTIES (A - E)
(A) a + b = b + a and ab = ba (commutative laws).
(B) (a + b) + c = a + (b + c) and (ab)c = a(bc) (associative laws).
(C) a(b + c) = ab + ac (distributive law).
(D) There are distinct real numbers 0 (additive identity) and 1 (multiplicative
identity) such that a + 0 = a and a1 = a for all a.
(E) For each a there is a real number −a such that a + (−a) = 0, and if
a NOT= 0, there is a real number 1/a such that a(1/a) = 1.
ORDER PROPERTIES (F - H)
(F) For each pair of real numbers a and b, exactly one of the following is true:
a = b, a < b, or b < a.
(G) If a < b and b < c, then a < c. (The relation < is transitive.)
(H) If a < b, then a + c < b + c for any c, and if 0 < c, then ac < bc
THEOREM: TRIANGLE INEQUALITY: If a and b are any two real numbers,
then |a + b| ≤ |a| + |b|.
PROOF:
There are four possibilities:
(a) If a ≥ 0 and b ≥ 0, then a + b ≥ 0, so |a + b| = a + b = |a| + |b|.
(b) If a ≤ 0 and b ≤ 0, then a + b ≤ 0, so |a + b| = −a + (−b) = |a| + |b|.
(c) If a ≥ 0 and b ≤ 0, then a + b = |a| − |b|.
(d) If a ≤ 0 and b ≥ 0, then a + b = −|a| + |b|.
DEFINITION: supremum of a set
A set S of real numbers is bounded above if there is a real number b such
that x ≤ b whenever x ∈ S. In this case, b is an upper bound of S. If b is an
upper bound of S, then so is any larger number, because of property (G). If
β is an upper bound of S, but no number less than β is, then β is a supremum
of S, and we write
β = sup S.
DEFINITION: infimum of a set
A set S of real numbers is bounded below if there is a real number a such
that x ≥ a whenever x ∈ S. In this case, a is a lower bound of S. If a is a
lower bound of S, so is any smaller number, because of property (G). If α is
a lower bound of S, but no number greater than α is, then α is an infimum of
S, and we write
α = inf S.
DEFINITION: the completeness axiom
If a non-empty set of real numbers is bounded above, then it has a supremum. (the real number system is a complete ordered field)
If a set S is unbounded above then sup S =
infinity
If a set S is unbounded below then inf S =
- infinity
The extended real number system is denoted by RBAR =
RBAR = [- infinity, infinity]
undefined forms such as infinity/infinity or 0/0 are called
intermediate forms
DEFINITION: subset of a set
Let S and T be sets.
S contains T, and we write S ⊃ T or T ⊂ S, if every member of T is
also in S. In this case, T is a subset of S.
DEFINITION: complement of a set
The complement of S, denoted by S^c, is the set of elements in the
universal set that are not in S
DEFINITION: singleton set
A set with only one member x0 is a singleton set, denoted by {x0}
DEFINITION: open interval
If a and b are in the extended reals and a < b, then the open interval (a, b)
is defined by
(a, b) = {x | a < x < b}.
DEFINITION: e-eighbourhood
If x0 is a real number and e > 0, then the open interval
(x0 − e, x0 + e) is an e-neighbourhood of x0
DEFINITION: interior point
If a set S contains an e-neighbourhood of x0, then S is a neighbourhood
of x0, and x0 is an interior point of S
DEFINITION: interior of a set
The set of interior points of S is the interior of S, denoted by S^0
DEFINTION: an open set
If every point of S is an interior point (that is, S
0 = S), then S is open
DEFINITION: a closed set
A set S is closed if S^c
is open.
DEFINITION: deleted neighbourhood
A deleted neighbourhood of a point x0 is a set that contains every point of
some neighbourhood of x0 except for x0 itself
THEOREM: (a) The union of open sets is open.
(b) The intersection of closed sets is closed.
These statements apply to arbitrary collections, finite or infinite, of open
and closed sets.
PROOF:
(a) Let G be a collection of open sets and S = ∪ {G | G ∈ G}.
If x0 ∈ S, then x0 ∈ G0 for some G0 in G, and since G0 is open, it contains
some e-neighbourhood of x0.
Since G0 ⊂ S, this e-neighbourhood is in S,
which is consequently a neighbourhood of x0. Thus, S is a neighbourhood of
each of its points, and therefore open, by definition.
(b) Let F be a collection of closed sets and T = ∩ {F | F ∈ F}. Then T^c = ∪ {F^c | F ∈ F} and, since each F^c
is open, T^c is open, from (a). Therefore, T is closed, by definition.
The intersection of finitely many open sets is …
open
The union of finitely many closed sets is …
closed
DEFINITION: limit point
Let S be a subset of R. x0 is a limit point of S if every deleted neighbourhood of x0 contains a point of S.
DEFINITION: boundary point
x0 is a boundary point of S if every neighbourhood of x0 contains at least one point in S and one not in S. The set of boundary points of S is the boundary of S, denoted by ∂S. The closure of S, denoted by S,
is S = S ∪ ∂S
DEFINITION: isolated point
x0 is an isolated point of S if x0 ∈ S and there is a neighbourhood of x0 that contains no other point of S
DEFINITION: exterior
x0 is exterior to S if x0 is in the interior of S^c. The collection of such points is the exterior of S
THEOREM: A set is closed if and only if no point of S^c is a limit point of S.
PROOF: Suppose that S is closed and x0 ∈ S^c. Since S^c is open, there is a
neighbourhood of x0 that is contained in S^c and therefore contains no points
of S.
Hence, x0 cannot be a limit point of S.
For the converse, if no point
of S^c is a limit point of S then every point in S^c must have a neighbourhood
contained in S^c. Therefore, S^c is open and S is closed.
DEFINITION: open covering
A collection H of open sets is an open covering of a set S if every point in S
is contained in a set H belonging to H; that is, if S ⊂ ∪ {H | H ∈ H}.
THEOREM: HEINE-BOREL THEOREM: If H is an open covering of a closed and bounded subset S of the real line, then S has an open covering H_TILDE consisting of finitely many open sets belonging to H.
PROOF:
DEFINITION: a compact set
a set that is both closed and bounded
THEOREM: BOLZANO-WEIERSTRASS THEOREM: Every bounded infinite set of real numbers has at least one limit point.
PROOF: We will show that a bounded nonempty set without a limit point can
contain only a finite number of points.
If S has no limit points, then S is closed and every point x of S has an open neighbourhood Nx
that contains no point of S other than x. The collection
H = {Nx | x ∈ S} is an open covering for S.
Since S is also bounded, S can be covered by a finite collection of sets from H, say Nx1 , . . . ,Nxn. Since these sets contain only x1, . . . , xn from S, it follows that S = {x1, . . . , xn}.
DEFINTION: if f(x) approaches the limit L as x approaches x0
lim(x→x0)f(x) = L, if f is defined on some deleted neighbourhood of x0 and, for every e > 0, there is a δ > 0 such that | f(x) − L | < e, if 0 < |x − x0| < δ.
THEOREM: If lim(x→x0)f(x) exists, then it is unique ; that is, if lim(x→x0)f(x) = L1 and lim(x→x0)f(x) = L2, then L1 = L2.
PROOF:
Suppose that the equations for L1 and L2 hold and let e > 0. By definition, there
are positive numbers δ1 and δ2 such that
|f(x) − Li| < e if 0 < |x − x0| < δi, i = 1, 2.
If δ = min(δ1, δ2), then
|L1 − L2| = |L1 − f(x) + f(x) − L2| ≤ |L1 − f(x)| + | f(x) − L2| < 2e if 0 < |x − x0| < δ.
We have now established an inequality that does not depend on x; that is,
|L1 − L2| < 2e.
Since this holds for any positive e, L1 = L2.
DEFINITION: f(x) approaches the left-hand limit
We say that f(x) approaches the left-hand limit L as x
approaches x0 from the left, and write lim(x→x0−)f(x) = L,
if f is defined on some open interval (a, x0) and, for each e > 0, there is a δ > 0 such that
| f(x) − L| < e if x0 − δ < x < x0.
DEFINITION: f(x) approaches the left-hand limit
We say that f(x) approaches the left-hand limit L as x
approaches x0 from the left, and write lim(x→x0−)f(x) = L,
if f is defined on some open interval (a, x0) and, for each e > 0, there is a δ > 0 such that
|f(x) − L| < e if x0 − δ < x < x0.
DEFINITION: f(x) approaches the right-hand limit
We say that f(x) approaches the right-hand limit L as x approaches x0 from the right, and write lim(x→x0+)f(x) = L,
if f is defined on some open interval (x0, b) and, for each e > 0, there is a δ > 0 such that
|f(x) − L| < e if x0 < x < x0 + δ.
DEFINITION: f(x) approaches the limit L as x approaches infinity
We say that f(x) approaches the limit L as x approaches
∞, and write lim(x→∞)f(x) = L,
if f is defined on an interval (a, ∞) and, for each e > 0, there is a number
β such that
| f(x) − L| < e if x > β.
DEFINITION: f(x) approaches infinity as x approaches x0 from the left
We say that f(x) approaches ∞ as x approaches x0 from
the left, and write
lim(x→x0−)f(x) = ∞ or f(x0−) = ∞,
if f is defined on an interval (a, x0) and, for each real number M, there is a δ > 0 such that
f(x) > M if x0 − δ < x < x0.
“limx→x0 f(x) exists” will mean that …
lim(x→x0)f(x) = L, where L is finite
DEFINITION: continuous at x0
We say that f is continuous at x0 if f is defined on an open interval (a, b) containing x0 and limx→x0 f(x) = f(x0).
DEFINITION: continuous from the left at x0
We say that f is continuous from the left at x0 if f is defined on an open interval (a, x0) and f(x0−) = f(x0).
DEFINITION: continuous from the right at x0
We say that f is continuous from the right at x0 if f is defined on an open interval (x0, b) and f(x0+) = f(x0).
DEFINITION: a function is continuous on an open interval (a,b)
A function f is continuous on an open interval (a, b) if it is continuous at every point in (a, b). If, in addition,
f(b−) = f(b) or f(a+) = f(a)
then f is continuous on (a, b] or [a, b), respectively. If f is continuous on (a, b) and both hold, then f is continuous on [a, b].
DEFINITION: f is a piecewise continuous function
A function f is piecewise continuous on [a, b] if
(a) f(x0+) exists for all x0 in [a, b);
(b) f(x0−) exists for all x0 in (a, b];
(c) f(x0+) = f(x0−) = f(x0) for all but finitely many points x0 in (a, b).
DEFINITION: composite function
Suppose that f and g are functions with domains Df and Dg. If Dg has a nonempty subset T such that g(x) ∈ Df whenever
x ∈ T, then the composite function f ◦ g is defined on T by (f ◦ g)(x) = f(g(x)).
THEOREM: Suppose that g is continuous at x0, g(x0) is an interior point of Df, and f is continuous at g(x0). Then f ◦ g is continuous at x0.
PROOF:
Suppose that e > 0. Since g(x0) is an interior point of Df and f is continuous at g(x0), there is a δ1 > 0 such that f(t) is defined and
| f(t) − f(g(x0))| < e if |t − g(x0)| < δ1.
Since g is continuous at x0, there is a δ > 0 such that g(x) is defined and
|g(x) − g(x0)| < δ1 if |x − x0| < δ.
Both conditions imply that
|f(g(x)) − f(g(x0))| < e if |x − x0| < δ.
Therefore, f ◦ g is continuous at x0.
DEFINITION: a function is bounded below
A function f is bounded below on a set S if there is a real number m such that f(x) ≥ m for all x ∈ S.
DEFINITION: a function is bounded above
A function f is bounded above on S if there is a real number M such that f(x) ≤ M for all x in S.
THEOREM: INTERMEDIATE VALUE THEOREM: Suppose that f is continuous on [a, b], f(a) NOT= f(b), and µ is between f(a) and f(b). Then
f(c) = µ for some c in (a, b).
PROOF: Suppose that f(a) < µ < f(b). The set S = {x | a ≤ x ≤ b}
and f(x) ≤ µ is bounded and nonempty. Let c = sup S.
We will show that f(c) = µ. If
f(c) > µ, then c > a and, since f is continuous at c, there is an e > 0 such that f(x) > µ if c − e < x ≤ c.
Therefore, c − e is an
upper bound for S, which contradicts the definition of c as the supremum of
S.
If f(c) < µ, then c < b and there is an e > 0 such that f(x) < µ for c ≤ x < c + e, so c is not an upper bound for S. This is also a contradiction.
Therefore, f(c) = µ.
DEFINITION: uniformly continuous function
A function f is uniformly continuous on a subset S of its domain if, for every e > 0, there is a δ > 0 such that
|f(x) − f(x’)| < e whenever |x − x’| < δ and x, x 0 ∈ S.
DEFINITION: a derivative of function f
A function f is differentiable at an interior point x0 of its domain if the difference quotient f(x) − f(x0) / x − x0, xNOT= x0, approaches a limit as x approaches x0, in which case the limit is called the derivative of f at x0, and is denoted by f'(x0); thus, f'(x0) = lim(x→x0)[f(x)−f(x0)]/[x−x0]. Usually let x = x0+h.
LEMMA: If f is differentiable at x0, then f(x) = f(x0) + f’(x0) + E(x), where E is defined on a neighbourhood of x0 and lim(x→x0)E(x) = E(x0) = 0.
PROOF:
DEFINITION: function f is differentiable on the closed interval [a,b]
We say that f is differentiable on the closed interval [a, b] if f is differentiable on the open interval (a, b) and f’+(a) and f’−(b) both exist.
DEFINITION: function f is continuously differentiable on [a,b]
We say that f is continuously differentiable on [a, b] if f is differentiable on [a, b], f’ is continuous on (a, b), f’+(a) = f’(a+), and
f’−(b) = f’(b−).
DEFINITION: local extreme value
We say that f(x0) is a local extreme value of f if there is a δ > 0 such that
f(x) − f(x0) does not change sign on (x0 − δ, x0 + δ) ∩ Df.
.
DEFINITION: local maximum/minimum value of function f
f(x0) is a local maximum value of f if f(x) ≤ f(x0)
or a local minimum value of f if f(x) ≥ f(x0) for all x in the set.
THEOREM: If f is differentiable at a local extreme point x0 ∈ D^0_f, then f’(x0) = 0.
PROOF:
We will show that x0 is not a local extreme point of f if f’(x0) NOT= 0.
f(x) − f(x0) / x − x0 = f’(x0) + E(x), where lim(x→x0)E(x) = 0.
Therefore, if f’(x0) NOT= 0, there is a δ > 0 such
that |E(x)| < |f’(x0)| if |x − x0| < δ,
and the right side must have the same sign as f’(x0) for |x − x0| < δ.
Since the same is true of the left side, f(x) − f(x0) must change sign in
every neighbourhood of x0 (since x − x0 does). Therefore, neither a local maximum or minimum can hold for all x in any interval about x0.
DEFINITION: critical point of function f
If f’(x0) = 0
THEOREM: ROLLE’S THEOREM: Suppose that f is continuous on the
closed interval [a, b] and differentiable on the open interval (a, b), and
f(a) = f(b). Then f’(c) = 0 for some c in the open interval (a, b).
PROOF: Since f is continuous on [a, b], f attains a maximum and a minimum value on [a, b]. If these two extreme values are the same, then f is constant on (a, b), so f’(x) = 0 for all x in (a, b).
If the extreme values differ, then at least one must be attained at some point c in the open interval (a, b), and f’(c) = 0.
THEOREM: INTERMEDIATE VALUE THEOREM: Suppose that f is differentiable on [a, b], f’(a) NOT= f’(b), and µ is between f’(a) and f’(b). Then f’(c) = µ for some c in (a, b).
Proof. Suppose first that
f’(a) < µ < f’(b) and define
g(x) = f(x) − µx.
Then g’(x) = f’(x) − µ, a ≤ x ≤ b,
and implies that g’(a) < 0 and g’(b) > 0.
Since g is continuous on [a,b], g attains a minimum at some point c in [a, b].
Lemma implies that there is a δ > 0 such that g(x) < g(a), a < x < a + δ,
and g(x) < g(b), b − δ < x < b, and therefore c NOT= a and c NOT= b.
Hence, a < c < b, and therefore g’(c) = 0 and f’(c)=µ.
The proof for the case where f’(b) < µ < f’(a) can be obtained by
applying this result to −f .
THEOREM: GENERALIZED MEAN VALUE THEOREM: If f and g are continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then [g(b)−g(a)]f’(c) = [f(b) − f(a)]g’(c)
for some c in (a, b).
PROOF: The function h(x) = [g(b)−g(a)]f(x)−[f(b)−f(a)]g(x) is continuous on [a, b] and differentiable on (a, b), and h(a) = h(b) = g(b)f(a) − f(b)g(a). Therefore, Rolle’s theorem implies that h'(c) = 0 for some c in (a, b). Since h'(c) = [g(b) − g(a)] f'(c) − [f(b) − f(a)]g'(c)
THEOREM: MEAN VALUE THEOREM: If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then
f’(c) = f(b) − f(a) / b − a
for some c in (a, b).
THEOREM: If f’(x)=0 for all x in (a, b) then f is constant on (a, b)
THEOREM: If f’ exists and does not change sign on (a, b), then f is monotonic on (a,b); either increasing, decreasing, non-increasing or non-decreasing.
THEOREM: MEAN VALUE THEOREM: If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then
f’(c) = f(b) − f(a) / b − a
for some c in (a, b).
THEOREM: If f’(x)=0 for all x in (a, b) then f is constant on (a, b)
THEOREM: If f’ exists and does not change sign on (a, b), then f is monotonic on (a,b); either increasing, decreasing, non-increasing or non-decreasing.
THEOREM: If |f’(x)| ≤ M, a
an infinite sequence of real numbers is a real-valued function defined on a set of integers …
{n | n ≥ k}
DEFINITION: a sequence converges to a limit s
A sequence {sn} converges to a limit s if for every e > 0
there is an integer N such that |sn − s| < e if n ≥ N. (in this case {sn} is convergent and lim(n→∞)sn = s)
DEFINITION: sequence diverging to +/-∞
We say that lim(n→∞)sn = ∞
if for any real number a, sn > a for large n or sn < a for large n for -∞
DEFINITION: {sn} diverges to ∞ if …
lim(n→∞)sn = ∞
DEFINITION: {sn} diverges to -∞ if …
lim(n→∞)sn = -∞
DEFINITION: a subsequence of a sequence
A sequence {tk} is a subsequence of a sequence {sn} if
tk = snk, k ≥ 0,
where {nk} is an increasing infinite sequence of integers in the domain
of {sn}. We denote the subsequence {tk} by {snk }.
THEOREM: If lim(n→∞)sn = s (−∞ ≤ s ≤ ∞),
then lim(k→∞)snk = s
for every subsequence {snk} of {sn}.PROOF: Consider the case where s is finite. For e > 0, there exist an integer N such that |sn − s| < e if n ≥ N. Since {nk} is an increasing sequence, there exists an integer K such that nk ≥ N if k ≥ K.
Therefore |snk − s| < e
if k ≥ K.
THEOREM: A point xBAR is a limit point of a set S if and only if there is a sequence {xn} of points in S such that xn NOT= xBAR for n ≥ 1, and
lim(n→∞) xn = xBAR.
PROOF: [IF] Suppose the sequence {xn} exists, then for all e > 0, there is an integer N such that 0 < |xn − x| < e if n ≥ N. Therefore every e-neighbourhood of xBAR contains infinitely many points of S, hence xBAR is a limit point of S.
[ONLY IF] Let xBAR be a limit point of S. Then for every integer n ≥ 1, the interval (xBAR - 1/n, xBAR + 1/n) contains some point xn in S with xn NOT= xBAR. Since |xm − xBAR| ≤ 1/n if m ≥ n, lim(n→∞)xn = xBAR.
THEOREM: If {xn} is bounded, then {xn} has a convergent subsequence.
PROOF: Let S be the set of distinct numbers of {sn}.
If S is finite, then there exists xBAR in S which occurs infinitely often in {xn} (i.e. there exists a subsequence {xnk} such that xnk = xBAR for all k). Then lim(k→∞) xnk = xBAR.
If S is infinite, then since S is bounded, the Bolzano-Weierstrass theorem implies that S has a limit point xBAR. There is a sequence of points {yj} in S, distinct from xBAR, such that lim(j→∞)yj = xBAR. However, {yj} may not be a subsequence of {xn} (i.e. it may not correspond to terms yj = xnj where {nj} is increasing). We can take an increasing subsequence of {nj} called {njk}. Then {yjk} = {snjk} is a subsequence of both {yj] and {xn}. Therefore this subsequence converges to xBAR.
DEFINITION: Cauchy sequence
A sequence of real numbers is said to be a Cauchy sequence if for any e > 0, there exists a N ∈ N such that if n ≥ N and m ≥ N, then |sn − sm| < ε.
LEMMA: Let {sn} be a convergent sequence of real numbers. Then {sn} is Cauchy.
PROOF: Suppose that sn→s as n→∞. Let ε > 0. Then there exists N such that if
n ≥ N, then |sn − s| < ε/2. Now suppose that n, m ≥ N, then |sn − sm|
= |(sn − s) − (sm − s)|
≤ |sn − s| + |sm − s| by the triangle inequality
< ε/2 + ε/2 = ε,
whence {sn} is Cauchy.
THEOREM: [Cauchy convergence criterion] Let {sn} be a Cauchy sequence. Then {sn} is convergent.
PROOF: Let {sn} be a Cauchy sequence. Then {sn} is bounded.
By the Theorem a), there is some convergent subsequence snk→s as k→∞, for some s ∈ R.
We claim that sk→s as
k→∞. Indeed, let ε > 0.
Then there exists N1 such that if k ≥ N1, then |snk − s| < ε/2.
But there exists also N2 such that if n, m ≥ N2 then
|sn − sm| < ε/2.
If k ≥ max(N1, N2) then nk ≥ N2 and nk ≥ N1 hence
|sk − s| = |(sk − snk) + (snk − s)|≤ |sk − snk| + |snk − s| < ε/2 + ε = ε
DEFINITION: infinite series
If {an}^∞_k is an infinite sequence of real numbers, the symbol ∞∑n=k a_n
is an infinite series, and a_n is the nth term of the series.
DEFINITION: converges to the sum A
We say that ∑∞n=k a_n converges to the sum A, and write ∞∑n=k a_n = A,
if the sequence {An}∞_k
defined by
An = ak + ak+1 + · · · + an,
n ≥ k,
converges to A. The finite sum An is the nth partial sum of ∑∞ n=k a_n.
If {An}∞_k diverges, we say that ∑∞ n=k a_n diverges; in particular, if
limn→∞ An = ∞ or −∞,
we say that ∑∞ n=k a_n diverges to ∞ or −∞.
A divergent infinite series that does not converge to +infinity is said to …
oscillate (or be oscillatory)
THEOREM: CUACHY’S CONVERGENCE CRITERION FOR SERIES: A series ∑ a_n
converges if and only if for every e > 0 there is an integer N such that
|an + an+1 + · · · + am| < e
if m ≥ n ≥ N.
PROOF: Let {A_n} denote the sequence of partial sums of ∑ an.
Then |Am − An−1| < e if m ≥ n ≥ N.
Since ∑ an is convergent, if and only if {An} is convergent.
This is equivalent to {An} being Cauchy.
COROLLARY: If ∑ an converges, then lim(n→∞) a_n = 0.
PROOF: Take m=n for (an + an+1 + · · · + am| < e if m ≥ n ≥ N.)
|an| < e if n ≥ N, that is lim(n→∞) a_n = 0.
COROLLARY: DIVERGENCE TEST: If lim(n→∞) a_n NOT= 0, then ∑ an diverges.
THE CONVERSE OF THIS IS NOT TRUE - COUNTEREXAMPLE FOR THIS IS THE HARMONIC SERIES.
DEFINITION: {Fn} converges pointwise on S to the limit function F
Suppose that {Fn} is a sequence of functions on D and the sequence of values {Fn(x)} converges for each x in some subset S of
D. Then we say that {Fn} converges pointwise on S to the limit function
F, defined by
F(x) = lim(n→∞)Fn(x), x ∈ S.
LEMMA: If g and h are defined on S, then
||g + h||S ≤ ||g||S + ||h||S (triangle inequality)
and
||gh||S ≤ ||g||S||h||S. (reversed triangle inequality)
notation means:
||g||S = sup(x∈S)|g(x)| = sup{|g(x)|x∈S}
DEFINITION: {Fn} converges uniformly to the limit function F on set S
A sequence {Fn} of functions defined on a set S converges uniformly to the limit function F on S if
lim(n→∞)||Fn − F||S = 0.
Thus, {Fn} converges uniformly to F on S if for each e > 0 there is an integer N such that ||Fn − F||S < e if n ≥ N.
THEOREM: CAUCHY’S UNIFORM CONVERGENCE CRITERION: A sequence of
functions {Fn} converges uniformly on a set S if and only if for each
e > 0 there is an integer N such that
||Fn − Fm||S < e if n, m ≥ N.
PROOF:
Suppose that {Fn} converges uniformly to F on S.
Then, if e > 0, there is an integer N such that
||Fk − F||S < e/2 if k ≥ N.
PROOF CONTINUED ON ELE.
THEOREM: If {Fn} converges uniformly to F on S and each Fn is continuous at a point x0 in S, then so is F. Similar statements hold for continuity from the right and left.
COROLLARY: If {Fn} converges uniformly to F on S and each Fn is continuous on S, then so is F; that is, a uniform limit of continuous functions is continuous.
DEFINITION: infinite series
If {fj}^∞k is a sequence of real-valued functions defined on a set D of reals, then ∑^∞(j=k)fj
is an infinite series (or simply a series) of
functions on D.
DEFINITION: partial sums of an infinite series (a series)
The partial sums of , ∑^∞_j=k(fj) are defined by
Fn = n∑(j=k)(fj), n ≥ k.
DEFINITION: converges pointwise to the sum F on set S
If {Fn}^∞_k
converges pointwise to a function F on a subset S of D, we say that ∑^∞_j=k(fj) converges pointwise to the sum F on S, and write
F = ∞∑j=k(fj), x ∈ S.
If {Fn} converges uniformly to F on S, we say that ∑^∞_j=k(fj) converges
uniformly to F on S.
THEOREM: CAUCHY’S UNIFORM CONVERGENCE CRITERION: A series ∑ fn
converges uniformly on a set S if and only if for each e > 0 there is an integer N such that
||fn + fn+1 + · · · + fm||S < e
if m ≥ n ≥ N.
COROLLARY: If ∑ fn converges uniformly on S, then lim(n→∞) ||fn||S = 0.
