The following question is from W. Cheney's book "Analysis for Applied Mathematics", Ex. 1.2, Problem 2:
Is this property of a sequence equivalent to the Cauchy sequence i.e. $\lim_{n\to\infty}\sup_{k\ge n}\|{x_{k}-x_{n}\|} = 0$
I find the above as true since the expansion of limits will result as $\sup_{k\geq N n\geq N}\|x_{k}-x_{n}\|<\epsilon$ for some $N$. Next the question proceeds as
Answer the same question for this property: For every positive $\epsilon$ there is a natural number $n$, such that $\|x_{m}-x_{n}\| < \epsilon$ whenever $m\geq n$
However, I find this as the same as above first part and equivalent since as in the first part one can define $N=\inf\{m:m\geq n\}$
Am I right?
For information, the book ( 2001 Edition) on Page 10 states the definition of a Cauchy sequence as
$[x_{n}]: \lim_{n \to \infty}\sup_{i\geq n j\geq n}\|x_{i}-x_{j}\| = 0$