Proving convergence of $|A| = \sqrt{A^*A}$ in the norm and strong operator topologies

Question

Let $\{A_n\}$ be a sequence of bounded linear operators on some Hilbert space, and let $|A| = \sqrt{A^*A}$. I would like to prove the following two statements:

1. If $A_n \rightarrow A$ is norm then $|A_n| \rightarrow |A|$ in norm.
2. If $A_n \rightarrow A$ strongly and $A_n^* \rightarrow A^*$ strongly, then $|A_n| \rightarrow |A|$ strongly.

If possible, I would like to do so without resorting to the spectral theorem. For both problems I am having some trouble showing that the square root of the operator preserves convergence. Any tips on how to solve this?