I think this identity is true but I can't get any formal proof (even though it seems to be quite easy):
If $1 < x \in \mathbb{R}$ and $ 1 \leq n \in \mathbb{N}$, then $x\lceil x^{n} \rceil \ge \lceil x^{n+1} \rceil - 1$.
I've done some simulations using Python and the identity holds for the bunch of numbers that I tried, but I'd like a formal proof of that.
If someone can help me I'd be very grateful.
Let $x^n=m-\delta$ with $0\le \delta<1$. Then $$\lceil x^{n+1}\rceil-x\lceil x^n\rceil =\lceil mx-\delta x\rceil-mx <(mx-\delta x+1)-mx\le 1$$