TITLE: Bounded away from 0 almost surely
QUESTION [0 upvotes]: Suppose that for a succession of random variables $X_n \to 1$ almost surely.
Does it hold that $P[X_n\geq 1/2]=1$ for $n$ large enough?

REPLY [4 votes]: No, in general that's not true. Consider for example the random variables $$X_n(x) := 1-x^n \qquad (x \in [0,1])$$ defined on the probability space $([0,1],\mathcal{B}([0,1]),\lambda|_{[0,1]})$ where $\lambda$ denotes the Lebesgue measure. Then $X_n \to 1$ almost surely, but $X_n(x) < \frac{1}{2}$ for all $x \in \left[\left( \frac{1}{2} \right)^{\frac{1}{n}},1\right]$.
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