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Let $f_1$ and $f_2$ be bounded functions on $[a, b]$. Prove that $$L(f_1) + L(f_2) \leq L(f_1 + f_2)$$
For this question specifically, I know that I am supposed to show that the
$$\sup L(f_1,p) + \sup L(f_2,p) \leq \sup L(f_1+f_2, p)$$
But I get confused when considering the sup of these three functions since the partition is any arbitrary partition.
I have also thought about trying to show $$\inf f_1 + \inf f_2 \leq \inf (f_1+f_2)$$ first in every interval, but I got stuck in terms of how to prove the above statement.
New to real analysis, would appreciate any insights.