The integer part of $x+1$ is the integer part of $x$ plus $1$

Question

How do you solve the proof:

If $x$ is a real number, then: $[x+1] = [x] + 1$.

For my proof, I tried to describe the interior of the argument inside the parentheses, but I was unsuccessful. Please help!

Answer 1

Let $y = [x] \Rightarrow x-1 <y \leq x\Rightarrow x < y+1 \leq x+1\rightarrow y+1 = [x+1]\Rightarrow [x]+1=[x+1]$