Proof for $\max (a_i + b_i) \leq \max a_i + \max b_i$ for $i=1..n$ I know this question is almost trivial because the truth of this statement is completely intuitive, but I'm looking for a nice and as formal as possible proof for $$\max (a_i + b_i) \leq \max a_i + \max b_i$$ with $i=1,\ldots,n$ 
Thanks in advance,
Federico

For any choice of $j$ in $1,2,\ldots,n$, we have that
$$a_j\leq\max\{a_i\}\quad\text{and}\quad b_j\leq\max\{b_i\}$$
by the very definition of "$\max$". Then, by additivity of inequalities, we have that for each $j$, 
$$a_j+b_j\leq\max\{a_i\}+\max\{b_i\}.$$
But because this is true for all $j$, it is true in particular for the largest out of the $a_j+b_j$'s; that is,
$$\max\{a_i + b_i\} \leq \max\{a_i\} + \max\{b_i\}$$
