Let be a Limit Point of C that is not in Y. (If there is no such point, we are done since C is then closed in X because C is closed in Y.) Since p is a Limit point there are infinitly many points of C in any neighbourhood of p. Since C is a subset of Y, there are also infinitely many points of Y in this neighbourhood and thus, p is also a limit point of Y. Since Y is closed in X, p must be in Y. This is a contradiction to our assumption. Thus any limit point of C must be in C if C is closed in Y and Y is closed in X.
If Y is not closed in X, e.g.:
(the real numbers) then C is closed in Y but Y is not closed in X. Also C is not closed in X.