For a real number $r$ we denote by $[r]$ the largest integer less than or equal to $r$.If $x,y \geq 1$ ,then which of the following statements is always true.
A) $[x+y] \leq [x]+[y]$
B) $[xy] \leq [x][y]$
C) $[2^x] \leq 2^{[x]}$
D) $[\frac xy] \leq |\frac xy|$
I tried the options by plugging in different values of x and y and till now I could not find x and y such that A) is incorrect.but that doesn’t guarantee that A) should be the correct option. I could not find out a pure algebraic proof of the same.please help me in this regard.thanks.
The statement $C$ is wrong! Try $x=1.9$.
$A$ is also wrong. Try $x=y=1.6$.
$B$ is also wrong: $x=y=1.5$.
$D$ is true of course.