Firstly, I'll give the definitions of sequential compactness and countable compactness.
Sequential compactness: If $X$ is a Hausdorff space and every sequence of points of $X$ has a convergent subsequence.
Countable compactness: if $X$ is a Hausdorff space and every infinite subset of $X$ has a cluster.
My text book gives me some counterexamples which are countably compact, even compact spaces, however they are not sequentially compact; we also can see this link compactness / sequentially compact. They are equal in first countable spaces.
But I think, without the condition of first countability, countable compactness implies sequential compactness. By the definition of countable compactness, every sequence of points of $X$ has a cluster point. Then this sequence has a subsequence (we choose itself) which is convergent, which shows that $X$ is sequentially compact. I don't know where I am wrong. Could anybody point out my mistakes? Thanks ahead:)