# If a sequence converges, and a subsequence converges to $x$, does the sequence converge to $x$?

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.

• Yes, it's true. Oct 24 '17 at 15:19

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.
If our sequence converges to $a$ then all subsequence converges to $a$.
Thus, $a=x$.