Definition 2.2.3 (Convergence of a Sequence)

A sequence (an) converges to a real number a if, for every positive number epsilon, there exists N in the natural numbers such that whenever n>=N it follows that |an-a|<epsilon

Definition 2.3.1 (Bounded Sequence)

A sequence (xn) is bounded if there exists a number M>0 such that |xn|<=M for all n in the natural numbers

Theorem 2.3.2

Every convergent sequence is bounded

Theorem 2.3.3 (Algebraic Limit Theorem)

Let lim(an)=a and lim(bn)=b. Then, (i) lim(can)=ca for all c in the real #s (ii) lim(an+bn)=a+b (iii) lim(anbn)=ab (iv) lim(an/bn)=a/b when b not =0

Theorem 2.3.4 (order limit theorem)

Assume lim(an)=a and lim(bn)=b (i) if an>=0 for all n in nat. #s, then a>=0 (ii) if an<=bn for all n in nat. #s, then a<=b (iii) if there exists c in real #s for which c<=bn for all n in nat. #s, then c<=b. Similarly, if an<=c for all n in nat. #s, then a<=c

Definition 2.4.1 (monotone sequence)

A sequence (an) is increasing if an<=an+1 for all n in nat. #s and decreasing if an>=an+1 for all n in nat. #s. A sequence is monotone if it is either increasing or decreasing

Theorem 2.4.2 (monotone convergence theorem)

If a sequence is monotone and bounded, then it converges

Definition 2.4.3 (convergence of a series)

Theorem 2.4.6 (Cauchy condensation test)

Definition 2.5.1

Let (an) be a sequence of real numbers, and let n1<n2<n3<... be an increasing sequence of natural numbers. Then the sequence (an1, an2, an3,...) is called a subsequence of (an) and is denoted (ank), where k in the nat. #s indexes the subsequence.

Theorem 2.5.2

Subsequences of a convergent sequence converge to the same limit as the original sequence

Theorem 2.5.5 (Bolzano-Weiestrass Theorem)

Every bounded sequence contains a convergent subsequence

Definition 2.6.1 (Cauchy sequence)

A sequence (an) is called a cauchy sequence if, for every epsilon>0, there exists an N in the nat. #s such that whenever m,n>=N it follows that |an-am|<epsilon

Lemma 2.6.3

Cauchy sequences are bounded

Theorem 2.6.4 (cauchy criterion)

A sequence converges is and only if it is a Cauchy sequence