Given a sequence in R is it sufficient to prove limit d(xn,xn+1) is 0 to show that it is cauchy?

Asked Jan 19 '19 at 06:03

Active Jan 19 '19 at 07:48

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I saw a proof in a fixed point theorem which showed that d(xn,xn+p) is 0 as n goes to infinity hence the sequence is cauchy. using triangle inequality if d(xn,xn+p) less than equal to d(xn,xn+1)+d(xn+1,xn+2)+....+d(xn+p-1,xn+p) And since each of these terms tend to 0 as n tends to infinity then that implies d(xn,xn+p) tends to 0 .Therefore,sequence is cauchy.

Am i missing a condition here? enter image description here

edited Jan 19 '19 at 07:48

asked Jan 19 '19 at 06:03

math123

2

What if $x_n=\frac11+\frac12+\frac13+\cdots+\frac1n$ ? – bof Jan 19 '19 at 06:07
Given a series $\sum_{n=1}^\infty a_n$ in $\Bbb R$, is it sufficient to prove that $\lim_{n\to\infty}a_n=0$ to show the series is convergent? – Angina Seng Jan 19 '19 at 06:13
One thing: $p$ could be related to $n$, like $p = n^2$. Then the sum is actually an infinite sum, and clearly the existence of such sum cannot be solely determined by the assumption $d(x_n, x_{n+1}) \to 0$. – xbh Jan 19 '19 at 06:46
No. See this. – David Mitra Jan 19 '19 at 06:48
For an Euclidean metric d(xn,xn+1) = xn + 1 - xn = 1. – William Elliot Jan 19 '19 at 08:58

Given a sequence in R is it sufficient to prove limit d(xn,xn+1) is 0 to show that it is cauchy?

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