Since this is too long for a comment, I post it as an answer:

Sorry, but I do not agree with Peter Michors answer. There are certainly better examples, but this the first I can remember: There are countable inductive limits $X=\lim\limits_{\to} X_n$ of Frechet spaces which are not Hausdorff (take a decreasing sequence of open connected sets $U_n$ in the complex plane with empty intersection and $X_n=H(U_n)$ the Frechet space of holomorphic functions on $U_n$ together with the injective restriction maps). $X$ is the a quotient of the direct sum $\bigoplus X_n$ which is certainly bornological and the kernel of the quotient map is sequentially closed
(because convergent sequences in the direct sum are located and convergent in some *finite* sum) but it is not closed because the quotient is not Hausdorff.

The situation is better for metrizable spaces (of course, this is trivial) as well as for so-called Silva spaces (also called LS or DFS-spaces, countable inductive limits of Banach spaces with compact inclusions): In these cases, sequentially closed *subspaces* are closed.