How to check convergence of sequence in complete metric space.

Question

Let $\{x_n\}$ and $\{y_n\}$ be two sequences in a complete metric space $(X,d)$ such that

1. $d(x_n,x_{n+1})\le\frac{1}{n^2}$
2. $d(y_n,y_{n+1})\le \frac{1}{n}$ , for all $n\in \mathbb{N}.$

Then which sequence would converge? Justify.

Now since the space is complete so every Cauchy sequence is convergent. Let $X=\mathbb{R}$ and $d$, the usual metric. So I take the sequence of partial sums of Harmonic series $\{H_n\}_{n\ge1}$, then

$$d(H_n,H_{n+1})=|H_{n+1}-H_n|=1/(n+1)\le1/n$$ But $H_n$ is not a Cauchy sequence so it does not converge in $X$.

So 2 need not converge.

Is my reasoning correct? Also I have a feeling that 1 would always converge, but I haven't be able to prove it. Can anyone help me with that? Thanks.

Answer 1

Your reasoning is good.

For 2, use the triangle inequality to show that, for $m < n$, $$d(x_n, x_m) \le \sum_{k = m}^{n - 1} d(x_k, x_{k+1}).$$ Cauchiness follows from the convergence of $\sum \frac{1}{n^2}$.