Suppose that the sequence $\displaystyle \{ x_n \} $ converges to a point x, prove that $\displaystyle x \in \ cl \{ x_1, x_2, x_3, . . . \} $

In addition, when would x be an accumlation point.

Proof so far.

So since $\displaystyle x_n \rightarrow x $, $\displaystyle \forall \epsilon > 0 $, there exist $\displaystyle N \in \mathbb {N} $ such that whenever $\displaystyle n \geq N $, we would have $\displaystyle d(x,x_n) < \epsilon $.

Now, I need to show that this x is in the closure, which is the intersection of all closed set containing $\displaystyle \{ x_1, x_2 , . . . , \} $.

So my idea is, for any closed set that contains the sequence, suppose that x is not in them, then x is not in the set that the sequence is in, then but the completeness property, that is impossible.

And x would be an accumlation point if the set is closed.

Is this right?