Suppose that $S = \{1,2,3,\cdots\}$ is the set of all positive integers and that $P(\{s\}) = 2^{-s}$ for all $s \in S$.
(a) Find a sequence of events $\{A_n\}$ such that $\{A_n\} \uparrow A_k$ where $A = \{2k+5 | k = 1, 2, 3 \}$
(b) Compute P(A), where $A = \{2k+5 | k = 1, 2, 3 \}$ by using continuity of P.
My attempt:
(a)
$$A_n = \{7, 9, 11, \cdots, 2n+5\}$$
(b)
Let $A_n = \{7, 9, 11, \cdots, 2n+5\}$. Then by finite additivity, $P(A_n) = P(7) + P(9) + \cdots + P(2n+5) = 2^{-7} + 2^{-9} + \cdots + 2^{-5 - 2n}$
$=1/4\frac{1-(1/4)^n}{1-1/4} = 1/3(1-(1/4)^n)$. But since $\{A_n\} \uparrow A$ we have $P(A) = \lim_{n\to\infty} P(A_n) = \lim_{n\to\infty} 1/3(1-(1/4)^n) = 1/3$.
Is this right?