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## Homework Statement

Suppose {Xn}, {Yn} are sequences in ℝ and that |Xn-Yn|→0. Show that either: a) {Xn} and {Yn} are both divergent or b) {Xn} and {Yn} have the same limit.

## Homework Equations

N/A

## The Attempt at a Solution

I first prove that lim(Xn-Yn)=lim(Xn)-lim(Yn). I am not completely clear on the validity of this proof.

Since |Xn-Yn|→0, |Xn-Yn-(x-y)|<ε. Now |Xn-Yn-(x-y)|=|(Xn-x)-(Yn-y)|≥|Xn-x|-|Yn-y| (from the triangle inequality).

So, |Xn-x|-|Yn-y|<ε. Therefore, the lim(Xn-Yn)=lim(Xn)-lim(Yn).

So, we have that lim(Xn-Yn)=lim(Xn)-lim(Yn). Then, if I assume Xn→x, then 0=x-lim(Yn). So, lim(Yn)=x. So, they have the same limit. If Xn does not have a limit, then both {Xn} and {Yn} must be divergent.

I would like to know if this is the correct approach to take and if the proof is correct. Thank You.