what would be an example of a convergent subsequence of an unbounded sequence?
$\displaystyle a_n = \left\{ {\begin{array}{lr}
{n,} & {n \text{ is even}} \\
{\frac{1}
{n},} & \text{otherwise} \\
\end{array} } \right.$
How do you prove this is unbounded? Is it enough if I prove that it is divergent and therefore unbounded? is there an easier way?
A sequence is really just a function defined on the natural numbers. And so you have a sequence $\displaystyle f:\mathbb{N}\mapsto\mathbb{Q}$ given by $\displaystyle f(n)=\begin{cases} n & \mbox{if} \quad n\text{ is even} \\ \frac{1}{n} & \mbox{if}\quad n\text{ is odd}\end{cases}$. A sequence is bounded then if $\displaystyle \text{diam}\text{ }f\left(\mathbb{N}\right)$ is bounded. This is clearly not the case here.