A metric space $X$ is closed if every convergent sequence $(x_n)$ in $X$ converges in the space. A space is closed if it contains all its boundary points, so then I thought one could show a space is closed by showing that an arbitrary point in the space is a boundary point. This could be done by taking an arbitrary point $x \in X$, and showing that for every $\epsilon > 0 $ one can find a point $y\notin X $, such that $d(x, y) < \epsilon$.
I understood this is an invalid way to show a space is closed, since with this method I managed to show that $c_{00}$ (the space of sequences thats eventually only zeroes) is closed. But $c_{00}$ is not closed, so my proof method above must clearly be wrong.
Where is my proof method wrong?