norm of bounded linear operator restricted to dense subspace I have no idea how to do this question I was given in class.
Let $E$ and $F$ be normed spaces and let $T \in \mathcal{L}(E,F)$. Suppose that $E_0 \subseteq E$ is a dense subspace. Show that $\parallel T_{E_0} \parallel = \parallel T \parallel$.
I know we can approximate every $x \in E$ by a sequence in $E_0$ but I can't see how to use it to get the result.
Any clues would be a big help

First note that, because its a subset of $E$,
$$||T||_{E_0} \leq ||T||$$
On the other hand given $\varepsilon >0$, there is some $x \in E$ such that $$||T|| \leq ||Tx|| + \varepsilon $$
And because $E_0$ is dense, and $T$ is continuous, there is an $y \in E_0$ near $x$ such that
$$||Tx|| - ||Ty|| \leq ||||Tx|| - ||Ty|||| \leq ||Tx - Ty|| < \varepsilon$$
Froms this we get
$$||T|| \leq ||Ty|| +2\varepsilon$$
We finish by letting $\varepsilon \rightarrow 0$ 
