91Ó°ÊÓ

Let \(\left(a_{n}\right)\) be an increasing sequence, \(\left(b_{n}\right)\) be a decreasing sequence, and assume that \(a_{n} \leq b_{n}\) for all \(n \in \mathbb{N}\). Show that \(\lim \left(a_{n}\right) \leq \lim \left(b_{n}\right)\), and thereby deduce the Nested Intervals Property 2.5.2 from the Monotone Convergence Theorem 3.3.2.

(a) Show that the series \(\sum_{n=1}^{\infty} \cos n\) is divergent. (b) Show that the series \(\sum_{n=1}^{\infty}(\cos n) / n^{2}\) is convergent.

Let \(A\) be an infinite subset of \(\mathbb{R}\) that is bounded above and let \(u:=\sup A\). Show there exists an increasing sequence \(\left(x_{n}\right)\) with \(x_{n} \in A\) for all \(n \in \mathbb{N}\) such that \(u=\lim \left(x_{n}\right)\)

Let \(\left(x_{n}\right)\) be a bounded sequence and for each \(n \in \mathbb{N}\) let \(s_{n}:=\sup \left\\{x_{k}: k \geq n\right\\}\) and \(S:=\inf \left\\{s_{n}\right\\}\). Show that there exists a subsequence of \(\left(x_{n}\right)\) that converges to \(S\).

Prove that if \(\lim \left(x_{n}\right)=x\) and if \(x>0\), then there exists a natural number \(M\) such that \(x_{n}>0\) for all \(n \geq M\).

Establish the convergence and find the limits of the following sequences. (a) \(\left((1+1 / n)^{n+1}\right)\), (b) \(\left((1+1 / n)^{2 n}\right)\), (c) \(\left(\left(1+\frac{1}{n+1}\right)^{n}\right)\), (d) \(\left((1-1 / n)^{n}\right)\).

Show that if \(\left(x_{n}\right)\) is unbounded, then there exists a subsequence \(\left(x_{n_{k}}\right)\) such that \(\lim \left(1 / x_{n_{k}}\right)=0\)

If \(\sum a_{n}\) with \(a_{n}>0\) is convergent, then is \(\sum \sqrt{a_{n}}\) always convergent? Either prove it or give a counterexample.

If \(x_{1}>0\) and \(x_{n+1}:=\left(2+x_{n}\right)^{-1}\) for \(n \geq 1\), show that \(\left(x_{n}\right)\) is a contractive sequence. Find the limit.

If \(\sum a_{n}\) with \(a_{n}>0\) is convergent, then is \(\sum \sqrt{a_{n} a_{n+1}}\) always convergent? Either prove it or give a counterexample.