Continuity from above

Topic: Probability spaces and events · more questions on this topic
‹ Two series of events

Prove continuity from above by applying continuity from below to the complements.

Worked answer (click to show)

If A_n\downarrow A, then A_n^c\uparrow A^c. Continuity from below gives \mathrm{P}(A_n^c)\uparrow\mathrm{P}(A^c). Subtract from 1:

\mathrm{P}(A_n)=1-\mathrm{P}(A_n^c)\downarrow 1-\mathrm{P}(A^c)=\mathrm{P}(A).