Let $(X,d)$ be a complete metric space, $r\in (0,1)$ and $(x_n)$ be a sequence in $X$ such that $d(x_{n+2},x_{n+1})≤rd(x_{n+1},x_n)$ for every $n\in N$. How can we show that $(x_n)$ is a convergent sequence?
$d(x_{n+2},x_{n+1})≤rd(x_{n+1},x_n)$ gives convergent sequence
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real-analysis
metric-spaces