This is an exercise in text R. Durrett, Probability: Theory and Examples, in the section entitled "Weak convergence".
Suppose $Y_n\geq0,EY_n^\alpha\rightarrow 1$ and $EY_n^\beta\rightarrow 1$ for some $0<\alpha<\beta$. Show that $Y_n\rightarrow 1$ in probability.
It seems reasonable that $Y_n$ is close to 1 for the following reason. We have $EY_n^\beta\geq(EY_n^\alpha)^{\beta/\alpha}$, because $x\to x^{\beta/\alpha}$ is convex. Here we actually have the "equality" when $n$ is big.
Maybe we should also link it to "weak convergence" in the proof since it's in that section... Please look at it for me. Thank you.