1
$\begingroup$

Say $a_n$ is a sequence and there exits two converging subsequences $a_{n_k}$ and $b_{n_k}$ such that $\lim_{n ->\infty}a_{n_k} \neq \lim_{n ->}b_{n_k}$. I need to prove that $a_n$ does not converge. In this question, I think proof by contradiction is a good approach. I suppose $a_n$ is a convergent sequence. Thus, if a sequence is convergent, then all of its subsequences are convergent and have the same limit, meaning that $\lim_{n ->\infty}a_{n_k} = \lim_{n ->}b_{n_k}$. There is a contradiction here.

Is my proof correct?

$\endgroup$
1
  • $\begingroup$ Well, that seems almost equivalent to what you are asked to prove. If you already have that theorem as something you can quote then, sure. But otherwise, I'd say you still need to add more detail. $\endgroup$
    – lulu
    Oct 2 '19 at 20:16
1
$\begingroup$

If you are given that "if a sequence is convergent, then all of its subsequences are convergent and have the same limit" your proof by contradiction is fine.

Otherwise you can start from the definition of limit for the sequence and for the two subsequences. Then choose $\epsilon_a$ and $\epsilon_n$ in such way that we obtain a contradiction for the limit of the sequence.

$\endgroup$
2
  • $\begingroup$ The statement 'if a sequence is convergent, then all of its subsequences are convergent and have the same limit' is not what I need to prove. In fact it's a theorem $\endgroup$ Oct 2 '19 at 20:36
  • $\begingroup$ @hck007 In that case your proof by contradiction is fine! $\endgroup$
    – user
    Oct 2 '19 at 20:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.