0
$\begingroup$

A few friends and I are doing a proof in our Real Analysis class, and we are wondering whether

If a sequence converges, and a subsequence converges to $x$, then the sequence also converges to $x$

is a true statement.

I know of a theorem that states "If a sequence converges to $x$, then every subsequence converges to $x$ as well." We are just wondering if the block-quote statement is valid.

Note: This isn't the proof we're working on, this is just a step in the middle of the proof.

$\endgroup$
1
  • 1
    $\begingroup$ Yes, it's true. $\endgroup$ Oct 24 '17 at 15:19
2
$\begingroup$

Yes, this is true.

The sequence converges, so suppose it does not converge to $x$, then it converges to a $y \neq x$. Now, apply the theorem you mention to conclude that the subsequence should converge to $y \neq x$, which is a contradiction.

$\endgroup$
1
  • $\begingroup$ Great, thanks for the help, your explanation is very clear. $\endgroup$
    – Smeef
    Oct 24 '17 at 15:24
1
$\begingroup$

Yes, it's true.

If our sequence converges to $a$ then all subsequence converges to $a$.

Thus, $a=x$.

$\endgroup$

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.