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Let $(x_n)$ and $(y_n)$ be Cauchy sequences in a metric space $(X, d)$. Show that the sequence $(d(x_n, y_n))$ is a cauchy sequence in $\mathbb{R}$.

What is the significance of $\mathbb{R}$ in this question? Anyway, I am trying to think about what the theorem would tell me. If $(d(x_n, y_n))$ is a Cauchy sequence then for large $n$ the distance between $x_n$ and $y_n$ must be $\epsilon$ close. Thus $x_n$ and $y_n$ must converge to the same point

Other than that I am lost.

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The significance of $\mathbb R$ in the question. That is the space where $d(x_n,y_n)$ lies. –  GEdgar Apr 17 '13 at 20:14
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up vote 1 down vote accepted

Hint By triangle inequality $$d(x_n,y_n)\leq d(x_n,x_m)+d(x_m,y_m)+d(y_m,y_n)$$ so $$d(x_n,y_n)-d(x_m,y_m)\leq d(x_n,x_m)+d(y_n,y_m)\tag{1}$$ and by the same method we have

$$d(x_m,y_m)-d(x_n,y_n)\leq d(x_n,x_m)+d(y_n,y_m)\tag{2}$$ hence with $(1)$ and $(2)$ we find $$|d(x_n,y_n)-d(x_m,y_m)|\leq d(x_n,x_m)+d(y_n,y_m)$$ Now use $(x_n)$ and $(y_n)$ are Cauchy sequences in a metric space $(X, d)$.

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I see that you are using the standard metric for $\mathbb{R}$ is this the significance with $\mathbb{R}$ in the problem? –  CodeKingPlusPlus Apr 17 '13 at 20:31
    
Yes that's exactly it. –  Sami Ben Romdhane Apr 17 '13 at 20:33
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