Real Analysis: show sequences have the same limit if |Xn-Yn| approaches 0

[nyr91188]

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.

 

[SammyS]

What you really need to prove are the following:

1) If one of them, say {x_n}, converges, then so does the other.

2) If one of them diverges, then so does the other.

You can alternatively prove this by contradiction as follows. Assume one of them converges, and one of them diverges. Then show that this,
together with lim_n→∞|x_n - y_n|=0, leads to a contradiction.

 

[nyr91188]

But you still have to prove that if both of them converge that they converge to the same value?
I don't quite understand what is implied by lim_n→∞|x_n-y_n|=0. Does this mean the lim(x_n)→lim(y_n)?

 

[SammyS]

But you still have to prove that if both of them converge that they converge to the same value?
I don't quite understand what is implied by lim_n→∞|x_n-y_n|=0. Does this mean the lim(x_n)→lim(y_n)?

No, you do.

lim_n→∞(x_n-y_n)=0 means |Xn-Yn|→0

 

[nyr91188]

I meant you as a general you, meaning whoever is doing the proof, meaning me. Anyway, I'm not looking for answers so calm down. I'm just trying to understand the steps of the proof. Does lim_n→∞(x_n-y_n)=0 mean that x_n and y_n converge to the same value (namely, that x=y)?

 

[nyr91188]

Also, I don't understand what would be wrong with proving lim(x_n-y_n)=lim(x_n)-lim(y_n) [which I now figured out how to prove correctly], and then say that assuming x_n→x then 0=x-lim(y_n) and lim(y_n)=x. Otherwise, if x_n does not approach x, then of course y_n is also divergent.

 

[Bacle]

nyr:

Assume for a moment that the difference x_n-y_n is always positive, or nonnegative. What can you then say about Lim_n→∞|x_n-y_n|? Consider then all other possible cases re the difference x_n-y_n