91Ó°ÊÓ

The sequence \(s_{n}=(-1)^{n}\) does not converge. For what values of \(\varepsilon>0\) is it nonetheless true that there is an integer \(N\) so that \(\left|s_{n}-1\right|<\varepsilon\) whenever \(n \geq N ?\) For what values of \(\varepsilon>0\) is it nonetheless true that there is an integer \(N\) so that \(\left|s_{n}-0\right|<\varepsilon\) whenever \(n \geq N ?\)

Give an example of a sequence that contains subsequences converging to every number in \([0,1]\) (and no other numbers).

Let \(\left\\{s_{n}\right\\}\) be a sequence that assumes only integer values. Under what conditions can such a sequence converge?

Determine all subsequential limit points of the sequence \(x_{n}=\cos n\).

Show that there cannot exist a sequence that contains subsequences converging to every number in \((0,1)\) and no other numbers.

A sequence \(\left\\{s_{n}\right\\}\) is said to be contractive if there is a positive number \(0

Let \(\left\\{s_{n}\right\\}\) be a sequence and obtain a new sequence (sometimes called the "tail" of the sequence) by writing $$ t_{n}=s_{M+n} \quad \text { for } n=1,2,3, \ldots $$ where \(M\) is some integer (perhaps large). Show that \(\left\\{s_{n}\right\\}\) converges if and only if \(\left\\{t_{n}\right\\}\) converges.

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 a sequence \(\left\\{a_{n}\right\\}\) has no convergent subsequences, what can you state about the lim sups and lim infs of the sequence?

Show that if \(\left\\{s_{n}\right\\}\) has no convergent subsequences, then \(\left|s_{n}\right| \rightarrow \infty\) as \(n \rightarrow \infty\)