Denote by $E(x)$ by the integer part of $x$, and let $x$ belong to the set of real positive numbers where $x$ doesn't equal to zero. Find $$\inf(E(x) +E(1/x)).$$ Firstly I tried plugging numbers for example 2 and it gave me 2, trying $1/2$ gave me 2, there is like a mirror happening here where numbers are split into 2 intervals that would yield the same answer, however between 1 and 2, excluding both numbers we would always get 1 and 1 is really the smallest element and so it is the infimum.
My question is how to prove it, I can not say that on a Real Analysis exam and thank you.