Question

Please show that, if x is a cluster point of {x^k}, and if d(x,x^k) >= d(x,x^(k+1)), for all k, then x is the limit of the sequence.

Answer 1

Question 1.

Show that, if $x$ is a cluster point of $\{x^{k}\}$, and if $d(x,x^{k})\geq d(x,x^{k+1})$, for all $k$, then $x$ is the limit of the sequence.

Solution. By definition of a cluster point for any $\varepsilon>0$ there is $K\in\mathbb{N}$ such that $0<d(x,x^{K})<\varepsilon$. Then for arbitrary $k>K$ we have

$d(x,x^{k})\leq d(x,x^{k-1})\leq\cdots\leq d(x,x^{K+1})\leq d(x,x^{K})<\varepsilon.$

So, for every $\varepsilon>0$ there is $K\in\mathbb{N}$ such that $d(x,x^{k})<\varepsilon$ for all $k\geq K$. This exactly means that $x$ is the limit of $x^{k}$. ∎