Unbounded sequence with convergent subsequence I'm just wondering if anyone knows any nice sequences that are unbounded themselves, but have one or more convergent sub-sequences?

There are plenty.
Take any convergent sequence, say $a_n \to a \in \mathbb R$. Then take any unbounded sequence, say $b_n \to \infty$. Then define $$ c_n = \begin{cases} a_n & \text{n even} \\ b_n & \text{n odd.} \end{cases}$$
Then $c_n$ is unbounded, but has a convergent sequence. Notice that you can generalize this: given any finite number of convergent sequences, you can make a unbounded sequence with the convergent sequences as subsequences.
