Question

prove that every convergent sequence is a cauchy sequence​

Answer 1

Answer: here is your answer

Step-by-step explanation:

Prove that if xn→a,n→∞ then {xn} is a Cauchy sequence.

I believe I have found the proof as follows, wondering if there are any simpler methods or added intuition. For me, it makes sense that if a sequence has a limit, then distances between elements in the sequence must be getting smaller, in order for it to converge.

Given ϵ>0,∃N1 s.t. ∀n≥N1:

|xn−a|<ϵ2<ϵ

and for m>n≥N1 we also have:

|xm−a|<ϵ2<ϵ

Let N≥N1, then ∀n,m≥N we have:

|xn−xm|=|xn−a−xm+a|<|xn−a|+|−(xm−a)|<ϵ2+ϵ2=ϵ

Therefore {xn} is a Cauchy sequence.

Also, if a sequence is Cauchy does it always converge? In other words, is it sufficient to check if a sequence is Cauchy to check for convergence