closed ball in euclidean space In general metric spaces the closed ball is not the closure of an open ball. 
However, I read that in the Euclidean space with usual metric, closed ball is the closure of an open ball. I'm having trouble rigorously proving this? How can I show this?

The only reason a closed ball might not be the closure of an open ball is when there is a "hole" between a point on the sphere and the centre.
Euclidean spaces have no "holes": if you take a point $x$ on a sphere of radius $r$ centred at a point $x_0$, then there is an interval connecting $x_0$ to $x$ and except $x$ it lies within an open ball of radius $r$.
The same argument works for normed spaces and general geodesic spaces, and with slight modification, for arbitrary length metric spaces.
