Limite con funzioni caratteristiche

[X-man1]

I am asked to show that
\begin{equation}\label{1}
f(s)=\frac{s^{2m}}{2m}\chi_{(0,\rho]}(s)+\left(\frac{\rho^{2\alpha}s^{2(m-\alpha)}}{2(m-\alpha)}-\frac{\rho^{2m}\alpha}{2m(m-\alpha)}\right)\chi_{(\rho,+\infty)}(s)<+\infty \quad \forall s>0
\end{equation}
where f is a real valued function, $0<\alpha0$ is a constant. Moreover, $\chi_{(a,b]}(s)$ is a characteristic function, that is, its value is $1$ for all $s\in(a,b]$ and $0$ otherwise.

This is true if and only if
$$\displaystyle\sup_{s>0}f(s)<+\infty
$$
I reckon this is false since taking $s>\rho$ we get
$$
\displaystyle\sup_{s>0}f(s)=\lim_{s\to+\infty}\left(\frac{\rho^{2\alpha}s^{2(m-\alpha)}}{2(m-\alpha)}-\frac{\rho^{2m}\alpha}{2m(m-\alpha)}\right)=+\infty
$$
This seems very trivial since $m-\alpha>0$ and therefore the exponent of s is positive, I wonder if I'm making a mistake.

[X-man1]

Grazie a chiunque ci abbiaa pensato, in realtà le due condizioni non son equivalenti, basta pensare alla funzione identità sui reali positivi. Pertanto la prima è verificata anche se la seconda non lo è.