"Definition: A sequence of real numbers (a(adsbygoogle = window.adsbygoogle || []).push({}); _{n}) isCauchyiff

for all ε>0, there exists N s.t. n≥N and m≥N => |a_{n}-a_{m}|<ε.

Anequivalentdefinition is:

for all ε>0, there exists N s.t. n≥N => |a_{n}-a_{N}|<ε. "

=============================================

I don't exactly see why these definitions are equivalent.

One direction (from 1st one to 2nd one) is clear, we can just take m=N which is clearly ≥N.

But how can we prove the converse (i.e. starting with the 2nd definition, prove the 1st)?

Any help is much appreciated!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Cauchy sequence

**Physics Forums | Science Articles, Homework Help, Discussion**