The contrapositive of a statement $A \to B$ is $\neg B \to \neg A.$
Let $(a_n) \subset \mathbb{R}$. In your case, $A$ is the statement $(a_n) \subset X$ and $B$ the statement $\exists (a_{n_i}): \lim_{i \to \infty} a_{n_i} = L \in X$. The negation of $A$ therefore is $(a_n) \nsubseteq X$ and the negation of $B$ is $\nexists (a_{n_i}): \lim_{i \to \infty} a_{n_i} = L \in X$. Therefore, the contrapositive of statement (b) is:
If for some sequence $(a_n) \subset \mathbb{R}$ there exists no subsequence which converges to some number $L$ in $X$, then $(a_n) \nsubseteq X$.
EDIT: After having read the original question again, I think we are actually trying to negate the whole statement (b). This can be done using the common rules of logic, noting that
$$(A \to B) \iff (\neg A \vee B).$$
Using this equivalence, the negation of an implication is given by
$$\neg(A \to B) \iff \neg(\neg A \vee B) \iff A \wedge \neg B.$$
The statements $A$ and $B$ are the same as in my original answer, and the negation of $B$ is given too.