Probability help needed

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#1
Use the fact that the mapping from subsets of R to [0,1] deﬁned by A 7→ P(X ∈ A) deﬁnes a probability measure on R, together with properties of probability measures to prove the following. If X is a random variable and (xn;n ≥ 1) an unbounded increasing sequence of real numbers then lim n→∞ P(X ≤ xn) = 1.

Could do with a hint on this please: it seems fairly obvious but not sure how to go about a proof.
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5 years ago
#2
Well, we're going to want to use some properties of probability measures (countable additivity always seems like a good bet). The first step is to define some sets A_n (there's a fairly obvious choice). Then we'll want to define some sets B_n in terms of the A_n which are disjoint.

Also, you can look at the 1 on the right-hand-side of the result we want. Where must it come from? It must come from our mapping P being a probability measure, and so P(the whole space) = P(R) = 1. Can we write R in terms of our sets defined above?
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