I could use some help understanding the following proof so please feel free to pitch in!

Consider any random variable \(X\) with distribution function \(F_X(x)\). Then given \(\epsilon >0\) we can bound \(X\) in the following way. Because the lower limit of \(F_X(x)\) is 0 and its upper limit is 1, we can find \(y_1\ and\ y_2\) such that:

\(F_X(x)<\epsilon/2\) for \(x\leq y_1\) and \(F_X(x)>1-\epsilon/2\) for \(x\geq y_2\)

Let \(y=max\left\{|y_1|,|y_2|\right\}\) then

\(\displaystyle P[|X|\leq y]=F_X(y)-F_X(-y-0)\geq 1-\epsilon/2-\epsilon/2=1-\epsilon\)

What I need help understanding is where the inequality in the last line comes from precisely. I know it has something to do with the bounds we set up above but I cannot figure it. I am a beginner so please explain in detail. Thanks.