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.
