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I am given this problem on a homework set for my analysis course. I'm honestly not sure what the question is asking, any hints would be greatly appreciated.

Does the sequence $$ a_n = \begin{cases} 1 & n=3k \\ 2 & n=3k+1 \\ 4 & n=3k+2 \end{cases}$$ converge? Prove your assertion.

I'm confused about the way the problem is written, I'm not sure how to interpret the sequence given.

Thank you.

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    $a_n$ has three subsequences that converge to different numbers. – 2017-02-21

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The sequence given is $1, 2, 4, 1, 2, 4, \ldots$

Can you see why this doesn't converge?

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    I do. But is it sufficient to say that as $k$ increases, the sequence will repeat as 1, 2, 4 indefinitely? Is using the $\epsilon$ notation ($|a_n - l|<\epsilon$ for converging sequences) necessary for this proof? – 2017-02-21
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    If you haven't proved that a sequence repeating 1, 2, and 4 indefinitely will not converge, then simply saying it repeats is not enough. And the epsilon-delta proof for this sequence is easy if proof by contradiction is OK. But you may have proved other facts about convergence that you can apply to the sequence with even less effort. It's really just a matter of what theorems you already have and can use; and we don't know what those are. – 2017-02-21
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Expanding on what DrMV commented on:

Theorem: If a sequence $\{a_n\}$ is convergent to $L$, then every subsequence $\{a_{n_k}\}$ converges to $L$ as well.

Contrapositive: If there exist two subsequences $\{a_{n_k}\}$ and $\{a_{n_m}\}$ that converge to $K$ and $M$, respectively, then $\{a_n\}$ cannot be convergent.

Your sequence has 3 distinct subsequences that converge to distinct values, so the original sequence cannot converge to any value.