By Jiri Lebl

A primary direction in mathematical research. Covers the genuine quantity process, sequences and sequence, non-stop features, the by-product, the Riemann necessary, sequences of features, and metric areas. initially built to educate Math 444 at collage of Illinois at Urbana-Champaign and later superior for Math 521 at college of Wisconsin-Madison. See http://www.jirka.org/ra/

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A common example of where monotone sequences arise is the following proposition. The proof is left as an exercise. 13. Let S ⊂ R be a nonempty bounded set. Then there exist monotone sequences {xn } and {yn } such that xn , yn ∈ S and sup S = lim xn n→∞ and inf S = lim yn . n→∞ 44 CHAPTER 2. 14. For a sequence {xn }, the K-tail (where K ∈ N) or just the tail of the sequence is the sequence starting at K + 1, usually written as {xn+K }∞ n=1 or {xn }∞ n=K+1 . The main result about the tail of a sequence is the following proposition.

3 3 3 48 CHAPTER 2. SEQUENCES AND SERIES Armed with this information we estimate |xn − x| = |xn − x + an − an | ≤ |xn − an | + |an − x| 2ε ε < + = ε. 3 3 And we are done. 2: A very simple example of how to use the squeeze lemma is to compute certain sequences using already known ones. For example, suppose that we have the sequence { n√1 n }. √ Since n ≥ 1 for all n ∈ N we have 1 1 0≤ √ ≤ . n n n for all n ∈ N. We already know that lim 1/n = 0. Hence, using the constant sequence {0} and the sequence {1/n} in the squeeze lemma, we conclude that 1 lim √ = 0.

Y| |y| And we are done. By plugging in constant sequences, we get several easy corollaries. If c ∈ R and {xn } is a convergent sequence, then for example lim cxn = c lim xn n→∞ and n→∞ lim (c + xn ) = c + lim xn . n→∞ n→∞ Similarly with subtraction and division. As we can take limits past multiplication we can show that lim xnk = (lim xn )k . That is, we can take limits past powers. Let’s see if we can do the same with roots. 6. Let {xn } be a convergent sequence such that xn ≥ 0. Then lim n→∞ √ xn = lim xn .