91Ó°ÊÓ

Show directly that the sequence \(s_{n}=1 / n\) is a Cauchy sequence.

On IQ tests one frequently encounters statements such as "what is the next term in the sequence \(3,1,4,1,5, \ldots ?\). In terms of our definition of a sequence is this correct usage? (By the way, what do you suppose the next term in the sequence might be?)

Show that every subsequence of a Cauchy sequence is Cauchy. (Do not use the fact that every Cauchy sequence is convergent.)

Suppose that \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a positive function with a derivative \(f^{\prime}\) that is everywhere continuous and negative. Apply Newton's method to obtain a sequence $$ x_{1}=\theta, \quad x_{n+1}=x_{n}-\frac{f\left(x_{n}\right)}{f^{\prime}\left(x_{n}\right)} $$ Show that \(x_{n} \rightarrow \infty\) for any starting value \(\theta\).

What relation, if any, can you state for the lim sups and lim infs of a sequence \(\left\\{a_{n}\right\\}\) and one of its subsequences \(\left\\{a_{n_{k}}\right\\} ?\)

If \(\left\\{s_{n}\right\\}\) is a sequence of positive numbers converging to 0 , show that \(\left\\{\sqrt{s_{n}}\right\\}\) also converges to zero.