Show that $[x] + [x + 1/2] =[2x]$, for all $x \in \mathbb{R}$, where $[ \cdot ]$ denotes the Greatest Integer Function.
I am having trouble formally showing this. Intuitively I understand that when the fractional component of $x$ is at least $0.5$ it'll add $1$ to both sides. As of right now I have $2$ inequalities:
$[x] \leq x < [x] + 1$
$[x + \frac12] \leq x + \frac12 < [x + \frac12] + 1$
I tried adding them together to form $[x] + [x + \frac12] \leq 2x + \frac12 < [x] + [x + \frac12] + 2$ in an attempt to rewrite this into a form for the floor function of $[2x]$ but am having some trouble. Is this the right track?