Suppose $P([\frac1n, \frac12]) \leq \frac13$ for all $n= 1,2,3,...$ Must we have $P((0, \frac12)) \leq \frac13$?

Question

Suppose $P([\frac1n, \frac12]) \leq \frac13$ for all $n= 1,2,3,...$

Must we have $P((0, \frac12)) \leq \frac13$?

This problem is from a Continuity (Boole's, Bonferroni's Theorems) section in a Statistics textbook. How do I prove this with a method that is relevant to the content taught in that section?

Thanks in advance!

Answer 1

$(0,\frac 1 2) \subset \cup_n [\frac 1 n, \frac 1 2]$ and the events $[\frac 1 n, \frac 1 2]$ increase as $n$ increases. Hence $P(0,\frac 1 2) \leq (\lim P([\frac 1 n, \frac 1 2])\leq \frac 1 3$.