Proving the Darboux Integral is Linear

Adapted from Alfonso-Gracia Saz, MAT137 at the University of Toronto. Today we prove rigorously that the Darboux Integral is linear. That is, if we want to integrate the sum of two functions, we can simply integrate the two individual functions and sum the result. This is an important result in Real Analysis.

00:00 Intro
1:13 Proof Header
2:17 Darboux Definition
3:38 Lemma 1
5:08 Lemma 2
6:37 Lemma 3
8:31 Lemma 4
9:59 Lemma 5
11:44 Lemma 6
12:44 Conclusion

[willassad]

5:51 -- M_i, f + M_i, g is an UPPER bound of h, and thus M_i, h must be the LEAST upper bound.

[MadScientyst]

I remember doing this in Real Analysis 2 & catching hell to understand it!
Fortunately my Lecturer studied at Waterloo in Canada & was a great expositor & Analyst.
The way u disseminated this proof reminded me of him & his pedagogical style...2 thumbs up man!