sequence in metric space with metric bounded by infinite sum

Question

Let $(b_n)_{n\in\mathbb{N}}$ be a sequence of positive real numbers such that $\sum_{n=0}^{\infty}b_n$ converges. Prove that if $(x_{n})_{n\in\mathbb{N}}$ is a sequence in a metric space $(X,d)$ which converges to $a\in X$, and $d(x_n,x_{n+1})\leq b_n$ for all $n\in\mathbb{N}$ then $d(x_n,a)\leq \sum_{k=n}^{\infty}b_k$ for all $n\in\mathbb{N}$.

How could I prove this?

Answer 1

Let $N \geq n$. Then repeated use of the triangle equality gives you \begin{align} d(x_n, x_N) &\leq \left( d(x_n, x_{n+1}) + d(x_{n+1}, x_N) \right) \ldots \leq \sum_{k=n}^{N-1} d(x_k,x_{k+1}) \leq \sum_{k=n}^{N-1} b_n. \end{align} Now let $N \to \infty$.