Since \([x] \le x\) and \([y] \le y\). Adding these two:
$$\begin{gather} x = [x] + \{x\} \\ y = [y] + \{y\} \\ xy = [x][y] + \{x\}[y] + [x]\{y\} + \{x\}\{y\} \end{gather} $$
Taking the floor of both sides:
$$ [xy] = [[x][y] + \{x\}[y] + [x]\{y\} + \{x\}\{y\}] $$
Since \([x][y]\) is an integer:
$$\begin{align} [xy] &= [x][y] + [\{x\}[y] + [x]\{y\} + \{x\}\{y\}] \\ [xy] &\ge [x][y] \end{align}$$