Two fundamental results in the theory of differentiable dynamical systems are the:
General density theorem:
For a $C^1$-generic diffeomorphism of a closed manifold, the periodic points are dense in the nonwandering set.
and the
Closing lemma:
Given a nonwandering point $p$ of a diffeomorphism $f$ of a closed manifold, one can find a $C^1$-small perturbation of $f$ so that $p$ is a periodic point.
I am not an expert in this field, but I normally think of the first theorem as of more intrinsic interest, and the second as a tool used to prove the first. But is it obvious that the first implies the second?
