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:


$$\Pr(\max\_{1 \le k \le n}{|S\_k|>\epsilon}) \le \\Pr(\max\_{1 \le k \le n/2}{|S\_k|>\epsilon/2}) + \Pr(\max\_{1 \le k \le n/2}{|M\_k|>\epsilon/2})$$




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**[added by YC]** for background context to this question, see [this MSE question](https://math.stackexchange.com/questions/94948/kolmogorovs-maximal-inequality-for-random-number-composition/95288#95288)



Help prove a maximal inequality
You can prove it by using the fact that the following holds always:


$\max\_{1 \le k \le n}|S\_k| \le \max\_{1 \le k \le n/2}|S\_k| + \max\_{1 \le k \le n/2}|M\_k|$


If the left hand side is larger than $\epsilon$ then one of the right hand terms is larger than $\epsilon/2$.


This also shows that the inequality is valid under absolutely no assumptions on the joint distribution of the variables $X\_1,\ldots,X\_n$.


