By Jiri Lebl

A primary path in mathematical research. Covers the true quantity procedure, sequences and sequence, non-stop services, the spinoff, the Riemann vital, sequences of features, and metric areas. initially constructed to coach Math 444 at collage of Illinois at Urbana-Champaign and later more desirable for Math 521 at collage of Wisconsin-Madison. See http://www.jirka.org/ra/

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13. 13: Let {xn } be a convergent monotone sequence. Suppose that there exists a k ∈ N such that lim xn = xk . n→∞ Show that xn = xk for all n ≥ k. 14: Find a convergent subsequence of the sequence {(−1)n }. 15: Let {xn } be a sequence defined by xn := n if n is odd, 1/n if n is even. a) Is the sequence bounded? (prove or disprove) b) Is there a convergent subsequence? If so, find it. 16: Let {xn } be a sequence. Suppose that there are two convergent subsequences {xni } and {xmi }. Suppose that lim xni = a and lim xmi = b, i→∞ i→∞ where a = b.

SEQUENCES AND SERIES We now get to the idea of a limit of a sequence. 6 that the notation below is well defined. That is, if a limit exists, then it is unique. So it makes sense to talk about the limit of a sequence. 2. A sequence {xn } is said to converge to a number x ∈ R, if for every ε > 0, there exists an M ∈ N such that |xn − x| < ε for all n ≥ M. The number x is said to be the limit of {xn }. We write lim xn := x. n→∞ A sequence that converges is said to be convergent. Otherwise, the sequence is said to be divergent.

We finish by the squeeze lemma. Yet another proof of the Bolzano-Weierstrass theorem proves the following claim, which is left as a challenging exercise. Claim: Every sequence has a monotone subsequence. 1: Suppose that {xn } is a bounded sequence. 1. Show that {an } and {bn } are bounded. 2: Suppose that {xn } is a bounded sequence. 1. Show that {bn } is an increasing sequence. 6. That is, suppose that {xn } is a bounded sequence and {xnk } is a subsequence. Prove lim inf xn ≤ lim inf xnk . 7.

### Basic Analysis: Introduction to Real Analysis by Jiri Lebl

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