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If we define a metric $d \colon \mathbf{R} \times \mathbf{R} \to \mathbf{R}$, where $\mathbf{R}$ is the set of real numbers, as follows: $$ d(x,y) := |\arctan x - \arctan y| $$ for all $x$, $y \in \mathbf{R}$, then how to determine if this metric space is complete?

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What is the range of $d$? –  Andres Caicedo Jan 28 '13 at 0:27
    
Note that the domain of $d$ is ${\bf R}\times{\bf R}$, not $\bf R$. –  Gerry Myerson Jan 28 '13 at 0:28
    
Exactly. I've incorporated the correction. –  Saaqib Mahmuud Jan 28 '13 at 0:53

1 Answer 1

HINT: $\langle n:n\in\Bbb N\rangle$ is a Cauchy sequence with respect to $d$.

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How to prove that the sequence $(n)$ is a Cauchy sequence in $\mathbf{R}$ under the metric in question? –  Saaqib Mahmuud Jan 28 '13 at 1:40
    
@Saaqib: Use the fact that $\lim_{n\to\infty}\arctan n=\frac{\pi}2$. –  Brian M. Scott Jan 28 '13 at 1:47
    
Is it possible to prove this limit statement rigorously? –  Saaqib Mahmuud Jan 31 '13 at 14:28
    
@Saaqib: Sure: $\tan x$ is an increasing function on $\left(-\frac{\pi}2,\frac{\pi}2\right)$, so $\arctan x$ is an increasing function on $\Bbb R$. –  Brian M. Scott Jan 31 '13 at 17:10

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