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

$$
   \max_{1\le i \le n} (a_i + b_i) \le \max_{1\le i \le n}( \max_{1\le k \le n}(a_k) + b_i) = \max_{1\le k \le n}(a_k) + \max_{1\le i \le n}(b_i)
$$
