Chapter 1 Flashcards
Definition 1.1 - Converging to the Limit L
Definition 1.1 - A real sequence (a_n) converges to the limit L if, for each epsilon>0, there exists N in the positive integers such that, for all n>_N, |a_n - L|< epsilon.
Proposition 1.4 - Uniqueness
The limit of a convergent sequence is unique.
Definition 1.5 - Bounded above/below
A sequence a_n is bounded above if there exists a real M such that for all n in the positive integers, a_n_< M. Any such M is called an upper bound on a_n.
A sequence a_n is bounded below if there exists a real K such that for all n in the positive integers, a_n >_ K. Any such K is called an lower bound on a_n.
A sequence is bounded if it is bounded above and below.
Proposition 1.6 - Boundedness
If a_n converges then a_n is bounded.
There exists a real K such that, for all n in the positive integers, |a_n|_<K.
Proposition 1.7 - Limits preserve non-strict inequalities
If a_n -> L and, for all n in the positive integers, K_< a_n < M, then K<L_<M.
Proposition 1.8 - The Squeeze Rule
Assume a_n_<b_n_<c_n, a_n -> L and c_n -> L. Then b_n -> L.
Proposition 1.9 - The algebra of limits
If a_n -> A and b_n -> B then:
a_n + b_n -> A+B
a_n*b_n -> AB
a_n/b_n ->A/B ,Provided B is not equal to zero.
Definition 1.10 - What is a subsequence?
b_k is a subsequence of a_n if there exists a strictly increasing sequence of positive integers n_k such that b_k = a_n_k.
Proposition 1.11 - Subsequence limits
if a_n -> L and b_k is a subsequence of a_n, then b_k -> L.
Theorem 1.14 - Montone Convergence Theorem
If a_n is bounded and monotonic, then a_n converges.
Definition 1.13 - Increasing/Decreasing/Monotonic
A sequence a_n is increasing if a_n+1>a_n for all n in the positive integers.
A sequence a_n is decreasing if a_n+1<a_n for all n in the positive integers.
It is monotonic if it is increasing or decreasing.
Theorem 1.15 - Bolzano-Weierstrass Theorem
Every bounded real sequence has a convergent subsequence.
Definition 1.16 - Cauchy
A real sequence a_n is Cauchy if, for each epsilon > 0, there exists N in the positive integers such that, for all n, m >_N, |a_n - a_m| < epsilon.
Lemma 1.17 - Convergence and Cauchy relation
If a_n converges then a_n is Cauchy.
Cauchy Stuff
If a_n is Cauchy then it is also bounded.
Let a_n be Cauchy, and assume some subsequence of a_n converges to L. Then a_n converges to L.
A real sequence converges if and only if it is Cauchy.
