Suppose $a_{n}$ is a converging increasing sequence (that is, that has a limit) and a limit is $\lim a_{n} = L$. Let ($b_{n}$) be another sequence that “interleaves” the first one, that is, such that $a_{n} <b_{n} < a_{n+1}$ for all n. Prove, using a limit definition, that $\lim b_{n} = L$ as well. Give an example that shows that the conclusion ($\lim b_{n} = L$) cannot be more valuable if $b_{n}$ not monotonous.
Any help guys?
Since we have the inequality that $a_n<b_n<a_{n+1}$, by the shift rule, $\lim_{n\to\infty}(a_n)=\lim_{n\to\infty}(a_{n+1})=L$, so by the Sandwhich Theorem $(b_n)$ converges, and $\lim_{n\to\infty}(b_n)=L$.
For an example: $a_n = (1-\frac{1}{2n})$, $b_n = (1-\frac{1}{2n+1})$
$(1-\frac{1}{2n})<(1-\frac{1}{2n+1})<(1-\frac{1}{2n+2})$,
$(a_n)\to 1$, and so $(b_n)\to 1$.