Graduate Constructing a sequence in a manifold

[JYM]

Given S is a submanifold of M such that every smooth function on S can be extended to a smooth function to a neighborhood W of S in M. I want to show that S is embedded submanifold.
My attempt: Suppose S is not embedded. Then there is a point p that is not contained in any slice chart. Since a submanifold is locally embedded, let U be a neighborhood of p that is embedded. Consider a smooth function on S that is supported in U and equal to 1 at p. If there is a sequence x_n in S-U that converges to p, then since f can be extended to a smooth function F on W but then 0=F(x_n) converges to F(p)=1, which is contradiction. My difficulty is to justify such a sequence exists. Please provide me your help.

 

[JYM]

No. We can construct such a sequence. Now I get the idea; the result follows from first countablity of manifolds ( as second countable is first countable).