You are on the right track:

Let { } be the convergent subsequence and suppose it converges to y. Then I claim 2 things:

1. f(y) = 0

2. f'(y) = 0

1. is easy and follows from continuity of f.

For 2, we use first principles:

But we know a) The limit exists (by assumption) and b) there's a sequence { } that converges to y and that f( ) = 0 for every point...