Let $E,F,G$ are normed spaces, $T\in\mathcal{L}(E,F), \ S\in\mathcal{L}(F,G)$. Prove that $S\circ T\in\mathcal{L}(E,G)$, where $\mathcal{L}(A,B)$ is a set of linear and continuous operators from $A$ to $B$.
The linearity is trivial, I would like to ask if my proof of continuity is valid. Even though there are some proofs of this (and more general), my goal was to understand if I manipulate properly with the notions of norms and continuity.
Proof: $$ S\in\mathcal{L}(G,F)\implies \forall \varepsilon>0\ \exists\delta>0: ||Tx_1-Tx_2||_F<\delta\implies ||S\circ Tx_1-S\circ Tx_2||_G<\varepsilon \ \ \forall Tx_1, Tx_2\in F \ \ (\forall x_1,x_2\in E, \forall T\in\mathcal{L}(E,F)) $$ Therefore $$ \forall\varepsilon>0 \ \exists\delta>0: ||T||_{\mathcal{L}(E,F)}||x_1-x_2||_E<\delta \implies ||S\circ Tx_1-S\circ Tx_2||_G<\varepsilon $$ Moreover, since $T$ is bounded, for every $\varepsilon$ we can choose $\frac{\delta}{||T||_{\mathcal{L}(E,F)}}$ and continuity condition will be satisfied.