= Solution
The <integer> endpoint hypothesis is important. Let $r_{n,k}=\binom nk f(k/n)-\lfloor\binom nk f(k/n)\rfloor$. The <floor function> gives $0\le r_{n,k}<1$, while $r_{n,0}=r_{n,n}=0$ because $f(0)$ and $f(1)$ are <integers>. Therefore the <rounded Bernstein polynomial> satisfies
$$
0\le B_n(f,x)-B_n^*(f,x)\le S_n(x):=\sum_{k=1}^{n-1}x^k(1-x)^{n-k}.
$$
For $0\le x\le1/4$, the successive terms have ratio $x/(1-x)\le1/3$, so the <geometric series> estimate gives
$$
S_n(x)\le\frac32x(1-x)^{n-1}\le\frac{3}{2n}.
$$
The last inequality follows by maximizing $x(1-x)^{n-1}$ at $x=1/n$. The same bound holds on $[3/4,1]$ by reflection. On $[1/4,3/4]$, each factor $x$ and $1-x$ is at most $3/4$, giving $S_n(x)\le(n-1)(3/4)^n$. Thus
$$
\boxed{\|B_n(f)-B_n^*(f)\|_\infty\le\max\left\{\frac{3}{2n},(n-1)(3/4)^n\right\}\longrightarrow0.}
$$
Here $n\ge2$; for $n=1$ the difference vanishes. Without the <integer> endpoint hypothesis the claim would fail: at $x=0$ the error is $f(0)-\lfloor f(0)\rfloor$, independently of $n$.
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