I want to prove that $$\operatorname{Var}\left(\sum\limits_{i=1}^m{X_i}\right) \leq m\sum\limits_{i=1}^m{\operatorname{Var}(X_i)} \,. \>$$
A too complicated proof is to write $$ a_{ij}=\sqrt {Cov(X_i,X_j)} \,, $$
$$\operatorname{Var}\left(\sum\limits_{i=1}^m{X_i}\right) = \sum a_{ij}^2 \leq \sum a_{ii}a_{jj} \leq \sum\sum a_{ii}^2$$
By Cauchy-Scwarz and then the permutation inequality.
I'm sure it can be shorter, but how?