Theoretical Test 1 Flashcards
Definition of Convergent Sequence
A sequence {xn} is said to converge to a number x in R, if for every e > 0, there exists an M in N such that |xn - x| < e for all n >= M. The number x is said to be the limit of {xn}.
Definition of a Subsequence
Let {xn} be a sequence. Let {ni} be a strictly increasing sequence of natural numbers (that is n1 < n2 < n3 < ···). The sequence {xni}
is called a subsequence of {xn}.
Definition of liminf and limsup
Let {xn} be a bounded sequence. Let an := sup{xk : k>= n} and bn := inf{xk : k >= n}. We note that the sequence {an} is bounded monotone decreasing and {bn} is bounded monotone increasing. Then
limsup xn := lim an
liminf xn := lim bn
Bolzano Weierstrass Theorem
Bolzano Weierstrass: every bounded sequence has a convergent subsequence
Definition of a Cauchy Sequence
A sequence {xn} is a Cauchy sequence if for every e > 0 there exists an M in N such that for all n >=M and all k >=M we have
|xn - xk| < e.
Theorem of Cauchy and Convergence
A sequence of real numbers is Cauchy if and only if it converges
Definition of a Series
Given a sequence {xn}, we write the formal object
Sum(xn). A series converges, if the sequence {sn} of partial sums converges
Definition of a Cauchy Series
A series Sum(xn) is said to be Cauchy or a Cauchy series, if the sequence of partial sums {sn} is a Cauchy sequence.
Proposition of Absolute Convergence
A series xn converges absolutely if the series |xn| converges. If a series
converges, but does not converge absolutely, we say it is conditionally convergent.
Definition of a Cluster Point
Let S ⇢ R be a set. A number x in R is called a cluster point of S if for every e > 0, the set (x - e , x + e ) union S \ {x} is not empty.
Definition of a Function Limit
Let f : S -> R be a function and c a cluster point of S. Suppose that there exists an L in R and for every e > 0, there exists a delta > 0 such that whenever x in S \ {c} and |x -c| < delta , then |f(x)-L|<e.
Lemma of Sequence relationship to Function Limit
Let S⇢R and c be a cluster point of S. Let f:S->R be a function.Then f(x)->L as x->c, if and only if for every sequence {xn} of numbers such that xn in S{c} for all n, and such that lim xn = c, we have that the sequence {f(xn)} converges to L.
Definition of Continuity
Let S ⇢ R, f : S->R be a function, and let c in S be a number. We say that f is continuous at c if for every e > 0 there is a delta > 0 such that whenever x in S and |x-c| < delta , then |f(x -f (c)| < e .
Min-Max Theorem
Let f : [a, b]->R be a continuous function. Then f achieves both an absolute minimum and an absolute maximum on [a, b].
Bolzano’s Intermediate Value Theorem
Let f : [a, b]->R be a continuous function. Suppose that there exists a y such that f(a) < y < f(b) or f(a) > y > f(b). Then there exists a c in [a,b] such that f(c) = y.
The theorem says that a continuous function on a closed interval achieves all the values between the values at the endpoints.
