Let $(X, d)$ be a complete metric space, $r∈ (0,1)$ 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 $n∈ℕ$. Show that $\{x_n\}$ is a convergent sequence.
Let $(X, d)$ be a complete metric space, $r∈ (0,1)$ 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 $n∈ℕ$. Show that $\{x_n\}$ is a convergent sequence.