Define $f(x):=x^{t-1}e^{-x}$. For $k=1,2,\dots$ let $$f_k(x)=\begin{cases}x^{t-1}\left(1-\frac xk\right)^k & 0<x<k\\0&k\le x\le \infty\end{cases}$$ Show that $f_k(x)\to f(x)$ and $f_k(x)\le f_{k+1}(x),\forall x>0$.
Hint: Use the arithmetic-geometric mean inequality. You may not use the definition of the exponential function as a limit.
I can show the first part without using the arithmetic-geometric mean inequality, but this is probably not the expected solution. I don't see any starting point for using the mean inequality.
How to prove this?
