I read the proof of the proposition "every cauchy sequence in a metric spaces is bounded" from(adsbygoogle = window.adsbygoogle || []).push({});

http://www.proofwiki.org/wiki/Every_Cauchy_Sequence_is_Bounded [Broken]

I don't understand that how we can take m=N[itex]_{1}[/itex] while m>N[itex]_{1}[/itex] ?

In fact i mean that in a metric space (A,d) can we say that

[[itex]\forall[/itex]m,n>N[itex]_{1}[/itex][itex]\Rightarrow[/itex] d(x[itex]_{n}[/itex],x[itex]_{m}[/itex])<1][itex]\Rightarrow[/itex][[itex]\forall[/itex]n[itex]\geq[/itex]N[itex]_{1}[/itex][itex]\Rightarrow[/itex] d(x[itex]_{n}[/itex],x_{[itex]_{N_{1}}[/itex]})<1]

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# Cauchy sequences

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