Prove if lim(xn+1/xn) ﹤ 1, then lim(xn) = 0 [ILIEKMATHPHYSICS]

This video references the book "Introduction to Real Analysis" by Bartle and Sherbert (Fourth Edition). This is given by Theorem 3.2.11 in the book.
Here are the preliminary results we used in this video:
Suppose (xn) is a sequence. If xn ≥ 0 for all n, then lim(xn) ≥ 0: • If xn ≥ 0 for all n, t...
Suppose 0 ﹤ b ﹤ 1. Then lim(b^n) = 0: • If 0 ﹤ b ﹤ 1 then lim(...
Thanks and enjoy the video!
Real Analysis (Bartle and Sherbert) Playlist: • Real Analysis (Bartle)...

[@Meghana_Nallamilli]

I like how you outline the proof so that this sketch can be generalised. Love your videos!

[@simoneoliveri9560]

Great content brother 🗿

[@up1663]

I believe this result is used in calculus for convergence of sequences all the time. Nice to see a proof of it.

[@gruntilda2]

Keep it up homie. I’m glad you’re so consistent bc it really helps me out sometimes, and it’s just cool stuff!

[@albertovelasquez9027]

I really needed this

[@RabinSaidÖsteränggymnasietNA1C]

Can't you figure out the fact that it's a geometric series and say x_(n+1)/x_n=L where 1>L>0. Then you say L(x_n)=x_(n+1) and since L is between 1 and 0 that means the function of x_n is decreasing exponentially and is approaching 0 as n--->infinite.