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.


marked as duplicate by Namaste, Adam Hughes, Shailesh, user91500, user223391 Jan 30 '17 at 7:34

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  • $\begingroup$ Hello and welcome to the site. One is encouraged to learn typesetting in LaTeX and mathjax. I tried to help you do this for this question. Please let me know if I got something wrong. Here is a link to a quick tutorial to the typesetting meta.math.stackexchange.com/questions/5020/… $\endgroup$ – mathreadler Jun 16 '16 at 21:49
  • $\begingroup$ many thanks! I will learn $\endgroup$ – lll Jun 16 '16 at 22:07