# Let $M$be a closed dense subspace of Hilbert space. Then prove that $M=H$

$$M$$ closed $$\Rightarrow$$ If $$u_n\in$$ M and $$u_n\to u$$ then $$u \in M$$

$$M$$ dense in $$H$$ $$\Rightarrow$$ $$\forall u \in H \ \exists u_n\in M: u_n\to u$$

$$M$$ subspace of $$H$$ $$\Rightarrow$$ $$M \subset$$ H

How I prove that $$M$$ is in fact the Hilbert space $$H$$?

On any topological space $$X$$, the only subset $$S$$ which is both closed and dense is $$X$$ itself. That's so because $$S=\overline S=X$$.
Let $$u\in H$$. Then, since $$M$$ is dense, there exists $$(u_n)\subset M$$ such that $$u_n\to u$$. Since $$M$$ is closed, $$u\in M$$. So, $$H\subset M\subset H$$, i.e., $$M = H$$.