Convergent subsequences converge to the same limit as the original sequence

Asked Apr 14 '22 at 11:08

Active Apr 14 '22 at 11:36

Viewed 108 times

I was studying the following problem which is nicely solved here. The problem is as follows:

Assume $(a_{n})$ is a bounded sequence such that every convergent subsequence of $(a_{n})$ converges to the same limit $a\in \mathbb{R}$. Show $(a_{n})$ must converge to $a$.

Now I am trying to see the necessity of $a_n$ to be bounded. Can you give me any counter-example when the above statement does not hold for a sequence that is not bounded?

edited Apr 14 '22 at 11:18

asked Apr 14 '22 at 11:08

Edward

1

If $a_n=n$, the hypothesis is vacuously satisfied. – David Mitra Apr 14 '22 at 11:13
2

What is the convergent subsequence of $a_n=n$? Do you think it is a valid counter-example? If there is no converging subsequence --> every convergent subsequence converges to the same limit? – Edward Apr 14 '22 at 11:17
2

A counterexample in the nonbounded case is given by $a_n = (1+(-1)^n)n$ (and this sequence has convergent subsequences) – TheSilverDoe Apr 14 '22 at 11:19
2

Now it makes things clear. Thanks @TheSilverDoe ! – Edward Apr 14 '22 at 11:22
2

@DavidMitra What do you mean ? All the convergent subsequences of $(a_n)$ converge to the same limit (which is $0$) in my example. – TheSilverDoe Apr 14 '22 at 11:28
1

@TheSilverDoe Oh, sorry... – David Mitra Apr 14 '22 at 11:35

Convergent subsequences converge to the same limit as the original sequence

0 Answers0