I proved the inequality below using Wald's identity and some tricky but easy manipulation, but I cannot do it using the suggestion from the source: "Hint: optional sampling!"
Here is the problem:
$X_1,X_2,\ldots$ are i.i.d. with $P(X_1>0)=1$. Let $S_n=\sum_{i=1}^nX_i$ and for $x>0$ define:
$$T_x=min\{n\geq1:S_n\geq x\}$$
Prove:
$$\frac12E[T_x]\leq\frac{x}{E[\min(X_1,x)]}\leq E[T_x]$$
My solution (shown here partially):
Rewrite as: $$x\leq E[\min(X_1,x)]E[T_x]\leq 2x$$
The middle expression is equal to $$E\left[\sum_{i=1}^{T_x}\min(X_i,x)\right]$$ by Wald's identity. Thus we could just prove:
$$x\leq \sum_{i=1}^{T_x}\min(X_i,x)\leq 2x,$$
which is not that hard.
Now how do they do it using optional sampling?