prove that $\lfloor x\rfloor\lfloor y\rfloor\le\lfloor xy\rfloor$ How can I prove that if $x$ and $y$ are positive then

$$\lfloor x\rfloor\lfloor y\rfloor\le\lfloor xy\rfloor$$


I am using the notation $\{x\}$ to mean the fractional part of $x$.
$$
x = \lfloor x \rfloor + \{x\}, \;\; y = \lfloor y \rfloor + \{y\} \\
xy = \lfloor x\rfloor \lfloor y \rfloor + \lfloor x\rfloor \{y\} + \{x\}\lfloor y\rfloor  + \{x\} \{y\} \\
\lfloor xy\rfloor=\lfloor x\rfloor \lfloor y\rfloor + \lfloor\lfloor x\rfloor\{y\}\rfloor + \lfloor\{x\}\lfloor y\rfloor\rfloor\\
\text{but } x\ge0,\;\;y\ge0\\
\lfloor\lfloor x\rfloor\{y\}\rfloor + \lfloor\{x\}\lfloor y\rfloor\rfloor\ge0\\
\therefore \lfloor xy\rfloor\ge\lfloor x\rfloor\lfloor y\rfloor$$
