Limits are unique

Limits are unique

In this video, I show that limits are unique. In other words, a convergent sequence must converge to exactly one number, it cannot have two different limits. And of course, what makes this so elegant is the proof, this is analysis at its finest, enjoy! :)

Other examples of limits can be seen in the playlist below.

[FT]

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

[sebmata]

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

[tmlen]

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.

[jayjayf]

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

[sebmata]

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

[late]

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)

[coreymonsta]

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

[heliocentric]

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

[late]

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