Two equivalent definitions of convergent sequences?

Question

I know that:

Definiton 1. The sequence $(x_n)$ in the metric space $(X,d)$ is said to converge to the point $x_0\in X$ if $$\forall\epsilon>0, \exists n_0\in\mathbb{N} \text{ such that } \forall n\geq n_0, d(x_n,x_0)<\epsilon.$$

In other words, the sequence $(x_n)$ in the metric space $(X,d)$ converges to the point $x_0\in X$ if $d(x_n,x_0)\rightarrow 0$ with $n\rightarrow\infty.$

Definiton 2. The sequence $(x_n)$ in the metric space $(X,d)$ converges to the point $x_0\in X$ if in every neighborhood $U_{x_0}$ of $x_0$ there exists a natural number $n_0$ such that it is satisfied $\forall n\geq n_0\Rightarrow x_n\in U_{x_0}.$

The point $x_0$ is said to be the limit of the sequence $(x_n)$. Write $x_n\rightarrow x_0, (n\rightarrow\infty)$ or $\lim_{n\to\infty}x_n=x_0.$

Can these two definitions be proved to be equivalent?

Answer 1

Where are you stuck? As some starting food for thought, notice that

1) For every open neighborhood $U_{x_0}$ of $x_0$, there exists an $\epsilon>0$ with $(x_0-\epsilon,x_0+\epsilon)\subset U_{x_0}$.

2) For every $\epsilon>0$, the set $(x_0-\epsilon,x_0+\epsilon)$ is an open neighborhood of $x_0$.

These two facts should suggest that both definitions have the same content.

Answer 2

Let $x_{n}$ converge to $x_{0}$ according to definition 1) and let $N$ be a neighbourhood of $x_{0}$.

Since we are working in a metric space this means exactly that some $\epsilon>0$ exists such that $d\left(x_{0},y\right)<\epsilon\Rightarrow y\in N$.

Then $n_{0}\in\mathbb{N}$ exists such that $n\geq n_{0}$ implies that $d(x_{0},x_{n})<\epsilon$ and consequently $x_{n}\in N$.

Conversely let $x_{n}$ converge to $x_{0}$ according to definition 2) .

Note that for every $\epsilon>0$ the set $N=\left\{ y\in X\mid d\left(x_{0},y\right)<\epsilon\right\} $ is a neighbourhood of $x_{0}$, hence some $n_{0}\in\mathbb{N}$ exists such that $n\geq n_{0}$ implies that $x_{n}\in N$ or equivalently $d(x_{0},x_{n})<\epsilon$.