# Incomplete Metric Space

• April 7th 2011, 04:20 AM
raed
Incomplete Metric Space
Dear Colleagues,

Could you please help me in solving the following problem:

show that the set of all real number constitutes an incomplete metric space if we choose
$d(x,y)=|arc tan(x) - arc tan(y)|$

Best Regards
• April 7th 2011, 05:27 AM
Opalg
Quote:

Originally Posted by raed
Dear Colleagues,

Could you please help me in solving the following problem:

show that the set of all real number constitutes an incomplete metric space if we choose
$d(x,y)=|arc tan(x) - arc tan(y)|$

Best Regards

What can you say about the sequence $(x_n)$, where $x_n = n$ ?
• April 7th 2011, 05:50 AM
raed
Thank you very much for your reply. But is this Cauchy.

Regards,

Raed.
• April 7th 2011, 05:54 AM
Opalg
Quote:

Originally Posted by raed
Thank you very much for your reply. But is this Cauchy.

That is what you have to decide, using the given metric. (Happy)
• April 7th 2011, 05:55 AM
Plato
Quote:

Originally Posted by raed
But is this Cauchy.

Here is a large hint: $\displaystyle\lim _{n \to \infty } \arctan (n) = \frac{\pi }{2}$.