Let $P$ be a probability on $R$. Prove that for every $\epsilon > 0$ there is a compact set $K$ such that $P(K) > 1 - \epsilon$.
First poster here, I read the H.W. FAQ, first course in non-measure based probability theory so only the most basic definitions are known.
Here are my thought thus far:
Let $F_p$ be the distribution function of $P$ such that: $F_p$ = $P((\infty, t])$
Because $F_p$ is increasing and right continuous, an interval $(a,b]$ must exist such that $P((a,b]) > 1- \epsilon$. (I am at a loss as to how to show this)
Then there is the issue of the interval being a compact set $K$. I have never learned about compact sets, from my understanding the interval $(a,b]$ is not a compact set, while the interval $[a,b]$ is a compact set.(Not sure about this either...)
Thanks for all your input!