Let $X_1,…,X_n$ are exchangeable of random variables, and $n$ is an even number. $S_k=X_1+\dots+X_k$. $M_k=X_{n/2}+\dots+X_{n/2+k}$.

I want to prove:

$$\Pr(\max_{1 \le k \le n}{|S_k|>\epsilon}) \le \\Pr(\max_{1 \le k \le n/2}{|S_k|>\epsilon/2}) + \Pr(\max_{1 \le k \le n/2}{|M_k|>\epsilon/2})$$

**[added by YC]** for background context to this question, see this MSE question