I have a problem with this excercise. I need your help.
Let $f:\mathbb{R}\longrightarrow\mathbb{R}$
$f(t)=t+[t]$
where $[\cdot]$ is the floor function.
Define the metric:
$d(x, y)=|f(x)-f(y)|\quad (x,y)\in\mathbb{R}^2$
The metric space $(\mathbb{R}, d)$ is complete?
Can $(\mathbb{R}, d)$ be expressed as countable union of its compact subset?
For 1. I think I have to show that a Cauchy sequence is convergent on $(\mathbb{R}, d)$ right?
For 2. I have no idea.
Help me please :)