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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.

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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