Show that $(x_n)_{n \in \mathbb{N}}$ is a convergent sequence in $R^\mathbb{N}$.

Question

Suppose that $(x_n)$ is a sequence satisfying $|| x_{n+1}-x_n || < 1/ 5^n$ for all $n \in \mathbb{N}$. Show that $(x_n)_{n \in \mathbb{N}}$ is a convergent sequence in $R^\mathbb{N}$.

At first my idea was to prove that $(x_n)_n \in \mathbb{N}$ it is from Cauchy. I need help please.

Answer 1

Yes, actually you know from here that $\sum x_{n+1}-x_n$ is normally convergent and so $\sum x_{n+1} -x_n$ is convergent, so $(x_n)$.