Let $\displaystyle f: X\rightarrow Y$ be a continuous map of topological spaces.

1) Let A and B be subsets of X such that the closure of A = closure of B. Prove that the closure of f(A) = closure of f(B).

2) Prove that if A is dense in X and f(X)is dense in Y then f(A) is dense in Y.

These both look fairly self-intuitive but how can we actaully prove them?