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Problem 326

Define a Cauchy sequence. Prove: a) In any metric space \(\mathrm{X}\), every convergent sequence is a Cauchy sequence. b) Suppose \(\left\\{p_{n}\right\\}\) is a Cauchy sequence in a compact metric space \(\mathrm{X}\), then \(\left\\{\mathrm{p}_{\mathrm{n}}\right\\}\) converges to some point of \(\mathrm{X}\)

Expert verified

A Cauchy sequence is a sequence \(\left\\{p_n\right\\}\) in a metric space \(X\) with distance function \(d\) such that, for any given \(\epsilon > 0\), there exists a natural number \(N\) with \(d\left(p_m, p_n\right) < \epsilon\) for all \(m,n \ge N\). (a) Every convergent sequence in a metric space is a Cauchy sequence, as proven using the triangle inequality and taking \(N = \max\{N_1, N_2\}\) for appropriate choices of \(N_1\) and \(N_2\). (b) If \(\left\\{p_n\right\\}\) is a Cauchy sequence in a compact metric space \(X\), then there exists a point \(p \in X\) such that \(\lim_{n \to \infty} p_n = p\), as proven by showing the existence of a convergent subsequence and applying the triangle inequality again.

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Chapter 11

Evaluate the limit $\lim _{\mathrm{x} \rightarrow \infty}\left[\left\\{\sqrt{(\log \mathrm{x}) \log (\log \mathrm{x})\\}} /\left\\{\mathrm{e}^{\sqrt{\mathrm{x}}}\right\\}\right]\right.$

Chapter 11

Find the upper and lower limits of the sequence \(\left\\{A_{n}\right\\}\) defined $$ \text { by } \mathrm{A}_{1}=0 ; \mathrm{A}_{2 \mathrm{~m}}=\left[\left(\mathrm{A}_{2 \mathrm{~m}-1}\right) /(2)\right] ; \mathrm{A}_{2 \mathrm{~m}+1}=(1 / 2)+\mathrm{A}_{2 \mathrm{~m}} $$

Chapter 11

Define a convergent sequence in a metric space \(\mathrm{X}\). Prove: a) \(\lim _{n \rightarrow \infty}\left(1 / n^{p}\right)=0\) for \(p>0\) b) \(\lim _{n \rightarrow \infty} n \sqrt{p}=1\) for \(p>0\) c) \(\lim _{n \rightarrow \infty} n \sqrt{n}=1\) d) $\lim _{n \rightarrow \infty}\left[\left\\{n^{\alpha}\right\\} /\left\\{(1+p)^{n}\right\\}\right]=0\( for \)p>0 . \alpha$ real. e) If \(|x|<1, \lim _{n \rightarrow \infty} x^{n}=0\)

Chapter 11

Show that the following sequences are convergent: a) $a_{n}=[\\{1 \cdot 3 \cdot 5 \ldots(2 n-1)\\} /\\{2 \cdot 4 \cdot 6 \ldots(2 n)\\}]$ \(\mathrm{n}=1,2,3, \ldots\) b) \(a_{n}=(1 / 1 !)+(1 / 2 !)+\ldots+(1 / n !)\) \(\mathrm{n}=1,2,3, \ldots\)

Chapter 11

Find the limit of the sequence defined by $$ \mathrm{x}_{1}=(2 / 3) $$ and $$ \mathrm{x}_{\mathrm{n}}+1=\left[\left\\{\mathrm{x}_{\mathrm{n}}+1\right\\} /\left\\{2 \mathrm{x}_{\mathrm{n}}+1\right\\}\right] $$

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