tag removed - Is it true that all the "irrational power" functions are almost polynomial ?

Hello all, the $Delta$ operator on functions $mathcal{N} to mathbb{R}$
(where $mathcal N$ denotes $lbrace 1,2, ldots , rbrace$ )defined by
$Delta(f)(n)=f(n+1)-f(n)$ is well-known and
it is not very hard to show by induction that
$f$ is a polynomial of degree $leq k$ iff $Delta^{k+1}(f)$ is identically zero, where
$Delta^{k+1}$ denotes $Delta$ iterated $k+1$ times. Now I say that
a function $f : mathcal{N} to mathbb{R}$ is "almost polynomial" iff
$Delta^{k}(f)$ is a bounded function for some $k$.

My question is : let $lambda >0$ be a non-integer, and let
$f(n)=n^{lambda}$. Is it true that $f$ is almost polynomial ?