Unbounded sequence with convergent subsequence

Question

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

Answer 1

If the sub-sequence can be finite, then any unbounded sequence will do.

If you want an unbounded sequence with a convergent infinite subsequence then take any unbounded sequence with a constant subsequence. E.g. $a_n = n \sin(\frac{\pi n}2)$.

Answer 2

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.

Answer 3

Here is an extreme example. Recall that the set of rationals is countably infinite. Enumerate the rationals as $r_1,r_2,r_3, \dots$. Then for every real number $x$, there is a subsequence of the sequence $(r_n)$ that has limit $x$.