I am asked to prove: If $\displaystyle $\displaystyle \lim_{n \rightarrow \infty} ||f_{n} - f|| = 0$$ then $\displaystyle $\displaystyle \lim_{n \rightarrow \infty} ||f_{n}|| = ||f||$$. I've been trying to answer this by an epsilon-delta argument, but I wonder if this is the best way. Should I try to do some algebra, converting $\displaystyle ||f_{n} - f||$ into an integral and then try to push the limit inside? If so, I'm not certain what permits me to move the limit inside a root.