Consider the normed space $(C([0,1]),||\cdot||_{\infty})$. Define $F:C([0,1])\to C([0,1])$ by
$F(f)=f^2$
Show that $F$ is continuous with respect to $||\cdot||_{\infty}$.
I've attempted to show this like so:
Let $f_{0}\in C([0,1])$. We want to show that $F$ is continuous at $F_{0}$.
Let $\epsilon>0$. Choose $\delta= ?$
Then for $f\in C$ such that $||f-f_{0}||_{\infty}<\delta$, we have: $||F(f)-F(f_{0})||_{\infty}\le ||f^2-f_{0}^2||_{\infty}\le ||(f-f_{0})(f+f_{0})||_{\infty}$
I seem to have hit a dead end, and don't even know if the last part is useful. Any tips?