I started studying functional analysis and I encountered an example of a metrizable space whose general picture I can't fully grasp.
Let $X$ be the set of all sequences of real numbers.
Let $x_n=\{{\xi_i^{(n)}}\}_{i=1}^{\infty}$ and let $x=\{{\xi_i}\}$. We say that $x_n \to x$ whenever $\xi_i^{(n)} \to \xi_i$ as $n \to \infty$ for every $i=0,1,2,3,...$
That way we get a non-metric space $s$. We'll show that the space $s$ can be metrized.
Let $x=\{{\xi_i}\}$ and $y=\{{\eta_i}\}$. We define
$$\rho(x,y)=\sum_{i=1}^{\infty}{\frac{1}{2^i}\frac{|\xi_i-\eta_i|}{1+|\xi_i-\eta_i|}}$$
The book then shows that the distance function satisfies the metric axioms and after that it shows that a sequence $x_n$ converges to $x$ iff the distance $\rho(x_n, x) \to 0$.
I don't quite understand the last part. Isn't it enough to show that the distance function satisfies the axioms for a metric space? Why do we need to show the two definitions of convergences are the same?
My understanding is that we sort of have a topological space with a defined convergence and we want our distance function $\rho$ to preserve the topology and be continuous.
Is this correct? What's the key idea behind this metrization thing?
