I'm not much informed about manifold but I should answer some questions about it. Based on the definition I have written an answer for the following question but I feel there is something wrong with it! Could you please help me? Q: Let M be a smooth manifold and suppose that we have an open cover for that. If S is a subset of M such that the intersection of each element of that cover and S is a submanifold, then S itself is a submanifold.
A: Fix an arbitrary point p in M. This point belongs to an element of that cover and as we know the intersection of that element with S is a submanifold, so there exists a map around p such that satisfies the condition, so we are done!
Yes, your proof is correct. The common way of expressing this idea is by saying "being a manifold is a local property".