Real Analysis Ep 11: Monotone convergence theorem

Episode 11 of my videos for my undergraduate Real Analysis course at Fairfield University. This is a recording of a live class.

This episode is about the monotone convergence theorem.

[quant-prep]

We need more analysis videos....given that you already taught in masters.
you made us fell in love with real analysis, hence now it is your duty to fullfil our desires.

[josemachado]

@chris staecker

I love your humility and motivation to present the material, not as something "that is simply difficult", but through the use of comments that express exactly what goes through the mind of any student who encounters the rigor of analysis for the first time. I admire your empathy. I feel sad about the lack of participation from students ( they are just why, maybe!). I bet you're a cool guy.

[TheTacticalDood]

That lim x_n = lim x_(n+1) trick was neat. Good job!

[jvf]

I get intuitively why taking the limit on both sides of the equation works (39:48), but what is a rigorous way of showing that it does? The way it's shown, it looks like playing around with notation. At one moment, x_(n+1) is a term, so a number, the next, it is a sequence.

[bansishah]

at 46:50, shouldn't the 1st term be undefined ? because it will be 1/(1*(1-1))

[boyinisuresh]

42:00 sir can u tell me y the limit value x=2 and not -1

[trestietumadur]

without guessing, how would you know that that sequence is bounded by 2?