Simple Probability Inequality with Stopping Times

Question

Suppose $U_1,...,U_n$ are independent random variable with $\mathbb{E}[U_i]=0$.

Define $Z_k:=\sum_{i=1}^k U_i$. Set $T:=\inf \lbrace k \in N \mid |Z_k|>2\alpha \rbrace$.

Clearly $\lbrace T=k \rbrace$ is independent of $(Z_n-Z_k)=\sum_{i=k+1}^n U_i$.

I do not get why:

$$\Pr(|Z_n|>\alpha)\geq \Pr(|Z_n-Z_T|\leq \alpha, T\leq n)$$

Answer 1

By the definition of $T$ we have $Z_T>2\alpha$ whenever $T<\infty$. If, in addition, $|Z_n-Z_T|\leq \alpha$ we get

$$|Z_n|-\alpha=\alpha + |Z_n|-2\alpha > \alpha + |Z_n|-|Z_T| \geq \alpha - |Z_n-Z_T|\geq 0.$$