Hash function, $h(k) = \lfloor km \rfloor$ is simple uniform for real $k$ independently, uniformly distributed in the range $0 \leq k < 1$
$h\in [m]^U$ satisfies the simple uniform hashing assumption if when $x\in U$ is chosen uniformly at random, then $h(x)$ is uniformly distributed over $[m]$, or equivalently $\forall i\in[m]: \Pr\limits_{x\in U}[h(x)=i]=\frac{1}{m}$. In our case we have:
$\Pr[h(x)=i]=\Pr\big[\lfloor mx \rfloor=i\big]=\Pr[i\le mx &lt; i+1]=\frac{i+1}{m}-\frac{i}{m}=\frac{1}{m}$.
We used the fact that if $x$ is uniformly distributed over $[0,1]$ then $\Pr[a\le
x\le b]=b-a$ (the equality holds with all four combinations of $\le$ and $&lt;$).