Proof that the linear map $T:\mathbb{R^n} \to\mathbb{R^m}$ is bounded I would just like to know if my following proof is correct:
Claim: If $T:\mathbb{R^n} \to\mathbb{R^m}$ is a linear map, then there exists $C > 0$ such that for every $x \in \mathbb{R^n}$$\|T(x)\| \le C\|x\|$.
Proof:  We have  $T(x) = \sum_{i=1}^{n}x_iT(e_i)$, so let $C = n\max(\|T(e_i\|)$.  Then,
$$\|\sum_{i=1}^{n}x_iT(e_i)\| \le \sum_{i=1}^{n}|x_i|\|T(e_i)\|$$
by the triangle inequality, and 
$$\sum_{i=1}^{n}|x_i|\|T(e_i)\| \le \sum_{k=1}^{n} \max|x_i|\max\|T(e_i)\| \le C\|x\|$$

Your proof looks fine. 
$%And now I avoid the black box bad answer detector just a bit.$
