Subsequences and convergence. Do all subsequences have to converge to the same limit for the sequence to be convergent?

Question

I was asked this question:

Prove that $a_n$ converges if and only if:

$a_{2n},a_{2n+1},a_{3n}$ all converge

I thought this was an easy generic question until I read the hint which said: Note: It is not required that the three sub-sequences have the same limit. This needs to be shown

This is what is confusing me because I have found two sources stating something different:

Proposition 4.2. A sequence an converges to L ∈ R if and only if every subsequence converges to L.

and

Let $a_n$ be a real sequence. If the subsequence $a_{2n}$ converges to a real number L and the subsequence $a_{2n+1}$ converges to the same number L, then $a_n$ converges to L as well.

So my question is: for a sequence $a_n$ to converge does it's subsequences have to converge to the same limit? (I suspect not) and if the answer is no can you help me prove why?

Thanks in advance

Answer 1

If a sequence has two subsequences that do not both converge to the same limit, then the sequence does not converge.

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This can be proven using the $\epsilon-\delta$ definition of convergence:

• Let $a_n$ converge to $L$, and let $\{a_{n_k}\}_{k\to\infty}$ be a subsequence of $a_n$.
• Let $\epsilon > 0$.
• Then, because $a_n$ converges to $L$, there exists some $N$ such that if $n>N$, then $|a_n - L|<\epsilon$.
• Because $n_k$ is an increasing sequence of integers (by definition of a subsequence), there exists such $K$ that $n_K > N$.
• Then, if $k > K$, we have $n_k > n_K > N$, and from the previous point, we get that $|a_{n_k} - L|<\epsilon$, meaning that $a_{n_k}$ converges to $L$.

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However, your case is special in that there is an overlap of sequences, for example $a_6$ is in the first and third sequence, and $a_9$ is in the second and third sequence. In fact, the third sequence alternates between the first two sequences, and you can use this to prove all three limits must be equal.

Answer 2

I think the word "required" in the hint is confusing. Focus instead on the last sentence:

Note: It is not required that the three sub-sequences have the same limit. This needs to be shown

In other words, you need to show that as a result of the given hypotheses, the three sub-sequences do have the same limit. You need to do that precisely because it is a necessary condition for $a_n$ to converge.