The paper I have posted a picture of is a paper of Feller. He shows that the functions $f_k$ are Laplace transforms of $C^\infty$ functions $u_k$. In order to execute his suggested proof, I actually needed to change the inductive hypothesis to "$u_k$ is $C^\infty$ and vanishes at $0$ while all of its derivatives vanish at $\infty$" which is true. The next sentence after the part I was able to capture in this screenshot then claims that the $u_k$ each have variation on $(0, x)$ that is $\le x$. He says he uses Schwarz' inequality to see this. I don't understand how. Can someone please tell me?