Limits are unique

[@FT029]

I like the contradiction end to the proof a lot more! Thanks for showing that.

[@dr.peyamsfan7158]

Analysis at its finest!!!!

[@tmlen845]

It seems the original statement lim ... = S already implies that this is true, because an = sign is used. And I general if a = b and a = c, it must be that b = c.

[@srpenguinbr]

Not exactly. It could mean the limit can take 2 different values. For instance, consider x=arcsin(0) you know x can be 0 or pi, so does that mean pi=0? or consider roots in complex numbers, they can take many different values and none of them is more correct than the others

[@orlandomoreno6168]

@@srpenguinbr But that's inconsistent notation. = is transitive. Sometimes notation is used informally like that when it means something else and it's inelegant.

[@martinepstein9826]

Good observation. Technically, before you prove limits are unique you shouldn't use that notation. You should just say s is _a_ limit and t is _a_ limit. Every time one expression "equals" two different values simultaneously baby Jesus cries ;(

[@sebmata135]

Thank you for the vid Dr. Peyam! I ran into the same question last night while I was attempting the chapter 9 homework.

[@leprofeet9989]

this was very well explained

[@jayjayf9699]

I’m glad you finally added a subscript to the N as in your other videos the notation is ambiguous

[@eliyasne9695]

This is quite elegant!

[@sebmata135]

Wow the argument at the end doesn’t rely on the Least Upper Bound axiom so it’s very powerful. You can use this theorem to prove Monotone Bounded Convergence => Least Upper Bound property

[@drpeyam]

Indeed! Check out the playlist, I do precisely that :)

[@heliocentric1756]

What if humans really don't have free will, how can we choose epsilon??

[@drpeyam]

Maybe epsilon chooses you 😂

[@jayasuryav8324]

what if w-w0 is episilon/2

[@coreymonsta7505]

I never thought of doing it like that! Could take epsilon as the average of s and t, and N as the max of the N1 and N2. Then past N, sn is always supposed to be in two disjoint intervals somehow. Even one instance of such a thing is crazy lol

[@late7245]

Is there any limit teorem that limit value would be equal to (s-t)/2 or (s+t)/2 ? (s,t mentioned in this video)

[@fullfungo]

Yes, the limit uniqueness, which is what this video is about. If S and T are as defined in the video, then (S+T)/2 is equal to S and is equal to T.

[@coreymonsta7505]

Could you clarify what your question is?

[@coreymonsta7505]

La Te my guess is that you’ve got the thumbnail picture in mind in some way?

[@hach1koko]

In the usual sense, no. But you can define other types of limits for which that could be the case. For example, the sequence u(n)=(-1)^n has alternating values 1,-1,1,-1 and so on, so in the usual sense it does not have a limit, but its "cesaro limit" is 0 which is the average of 1 and -1. (the cesaro limit is the limit of (1/(n+1))*(sum u_k from k=0 to n), it's the limit of the average of the nth firsts u_k terms). One can prove that if a sequence has a limit L, then its cesaro limit is also L. So it's a more "general" definition of a limit.

[@mikhailmikhailov8781]

R is Hausdorff. Done.

[@mikhailmikhailov8781]

I never liked doing things in the style of real analysis, when there is a visual way to understand the result that works very generally. Limits and continuity are just that.

[@late7245]

Please analyze lim n->inf (sin(arcsin(n)) in another episode?

[@snejpu2508]

Arcsin is inverse of sin, so sin(arcsin(n)) is just n.

[@late7245]

@@snejpu2508 nope.graph( sin(arcsin(x))) != graph (x). For this reason limit must be analyzed.

[@snejpu2508]

@@late7245 Oh, right. It works only for x from -1 to 1. So actually the x cannot approach infinity, because it's not in the domain. It's like calculating the limit of x approaching minus infinity of sqrt(x). I'm not sure, if we can even do that.

[@late7245]

Pardon. I meant arcsin(sin(n)) ?

[@iamtrash288]

arcsin takes only the values from -1 to 1 on the real number line though

[@Domzies]

I knew we should have used epsilon/2