"Given a metric space (X,ρ), define a new metric on X by σ(x,y)=min{ρ(x,y),1}. The reader can check that σ is indeed a metric on X.(adsbygoogle = window.adsbygoogle || []).push({});

Claim 1: x_{n}->x in (X,ρ) iff x_{n}->x in (X,σ).

Claim 2: {x_{n}} is Cauchy in (X,ρ) iff it is Cauchy in (X,σ). (Hence completeness is the same for these two metrics.)"

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Right now, I'm trying to prove claim 1.

x_{n}->x in (X,ρ) means:

for all ε>0, there exists N s.t. if n>N, then ρ(x_{n},x)<ε

x_{n}->x in (X,σ) means:

for all ε>0, there exists M s.t. if n>M, then σ(x_{n},x)<ε

Nowif 0<ε<1, then ρ(x_{n},x)<ε <=> σ(x_{n},x)<ε, so taking N=M above works in the definition of limit. Am I correct so far?

But if ε>1, I think it is NOT true that ρ(x_{n},x)<ε <=> σ(x_{n},x)<ε, right? So what should I take N or M to be in the case whenε>1? What should I do for this case?

Thanks for any help!

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