Intro to Real Analysis (aka 'Baby' Real Analysis). This video shows a proof that sup(A+B) = sup(A)+sup(B)
Texts/references:
1) Real Analysis: A Long-Form Mathematics Textbook by Jay Cummings
2) Introduction to Real Analysis 4th ed. by Bartle & Sherbert
3) Principles of Mathematical Analysis 3rd Ed. by Rudin (aka 'Baby Rudin')
Timestamps
0:00 What we're going to prove
0:42 Discussion using 'simple' examples
2:05 Strategy for the proof
3:25 Proof of sup(A+B)=sup(A)+sup(B)
20:40 Reviewing case 2 of the proof