Proving All the Sequence Limit Laws | Real Analysis

We'll prove the five basic limit laws for convergent sequences. The limit laws for the sum of convergent sequences, difference of convergent sequences, constant multiples of convergent sequences, product of convergent sequences, and the quotient of convergent sequences. There are, of course, other limit laws not proven here, but these are typically presented as "the sequence limit laws" or "algebraic limit theorems". They are our first batch of general laws for manipulating convergent sequences and their limits.

TABLE OF CONTENTS
0:00 Motivating Example and Intro
3:37 Sum of Sequences
10:39 Difference of Sequences
17:07 Constant Multiple of a Sequence
22:18 Product of Sequences
36:01 Quotient of Sequences
51:52 Using the Limit Laws to Make a Nasty Example Easy as Pie

Let a_n converge to a, b_n converge to b, and let c be a real number. The limit laws we prove are as follows...

a_n+b_n = a+b
a_n - b_n = a-b
c*a_n = c*a
a_n*b_n = a*b
a_n/b_n = a/b provided b isn't 0 and each b_n isn't 0

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[ndafapheinrich]

Honestly, all the confusions and stress about these theorems proofs, that was in my mind got washed out by your teaching 😊😊😊....keep it high my

[adoxographer]

I can't thank you enough for these videos. You take the mystery out of proof writing in a way I have not seen anywhere else.

[vh]

6:48 this guy is wayyy ahead of his time

[dieselguitar]

I hear real analysis is known for being a very difficult course. I'm glad to have the opportunity to begin learning it with these detailed explanations before ever doing an analysis class. I'm gonna be so far ahead of my classmates (assuming I get accepted).

[tshepangjohnmolatedi]

I cant tell you how insightful your videos have been🔥before i was trying so hard to have the defintions down but now I can literally write a proof without thinking twice

[CantTouchMe]

It feels so good to have so many proofs in video format. Thank you man!

[SacredSphynx]

Honestly Ty so much, you are saving me from my uni lectures, i literally only know whats going on because of ur videos. Thanks so much. Wish u were my lecturer

[Dupamine]

Today i was feeling good for almost completing 30 videos . I have been studying this playlist for 2 months now. But you had to drop this 50 minute video haha ! This will be tough!

[akihayakawa]

soooo helpful i’d much rather watch these vids than read my lecturers notes lol u explain it more clearly tysm!!!

[christinesustraido]

The best explanation I ever heard. Thank you!!

[alishash]

Hey! Thanks for the video. It was insightful. ❤ I have a question regarding it. At 30:13, we added 1 to the 2c to make sure if c was 0, we didn't face the division by zero problem. However, can we say c could be -1/2, thus 2c = -1 and 2c + 1 = 0? And, we face the division by zero problem again... Is this right? And, how can we make sure it doesn't happen? Thank you again. ❤

[okikiolaotitoloju]

Quick question how would you prove the limit law for {an^p} -> a^p?

[cahlilthomas]

Hey, thanks for the videos. They're much appreciated. When you say "we can make the absolute value of |a_n - a| as small as we want", what exactly do you mean by that?

[clydefrog]

At 27:09 is it possible to just say since |a| is a constant, it converges by def as the lim(n->inf) a = a so then just claim |a| < 1/e and also have |bn-b|<e^2/2 I feel like that would be easier no? Thanks for the video :) edit: just noticed someone already made the comment, here I thought I was clever :(

[begum]

Thank you very much for the detailed explanations of the proofs. It helped a lot.

[Jewisaosj]

when dealing with functions, is delta also have to be the max of delta 1 and delta 2? or minimum of delta 1 and 2?

[SimchaWaldman]

26:16 Here I choose an L > 0 such that both |a| < L and |bₙ| < L.

[percy]

That was really helpful, thanks
Love from India ❤️

[imeldatorres]

Thank you very much for the videos you create. I am wondering if you can help me with this question. Determine if the sequence is convergent of divergent. ( 2+ (-1)^n)/n

[ralph]

brilliant explanation. saving my January exams