Suppose P([0,∞)) = 1. Prove that there is some n such that P([0,n]) > 0.9
I cant figure this out.
You are given that $\displaystyle \lim _{n \to \infty } P\left( {\left[ {0,n} \right]} \right) = 1$.
That means that $\displaystyle \left( {\exists N} \right)\left[ {n \geqslant N\, \Rightarrow \,\left| {P\left( {\left[ {0,n} \right]} \right) - 1} \right| < 0.1} \right]$.
From there $\displaystyle 1 - 0.1 < P\left( {\left[ {0,N} \right]} \right)$.