Thread: Limit Proofs (Sup and Inf)

please show and explain
a. B = lim sup An = {w : w element An for infinite many values of n}
and C = lim inf An {w:w element An for all but finitely many values of n}


b. Let (X, F, P) be the probabilty space with X = (0, 1), F the Borel sets and P the uniform distribution. Let An = (1/2 + 1/n, 1 - (1/n))
Calculate Cn, Bn, lim sup n--> infite An and lim inf n --> infinity An. Does A = lim n--> infitite exist? If so, what is P(A)?


c. Let An = (0,(1/2) - (1-n)) if n is odd
((1/2), 1 - (1-n) if n is even

again calculate Cn, Bn, lim sup n -->inifinity An and lim inf n-->infinity An. Does lim n --> infinity An exist? If so, what is P(A)?

For a), you need to define the sequence before you can draw any conclusions about it.

For b) and c), you need to define the sequences and before you can calculate them.

a. The hint given is

w element of "intersection"(m=n to infinite) "union"(m=n to infinite) Am
---> w element "union"(m=n to infinite) Am for all m
n >N w element of An

hence how do you you proof this backwards now?


ohh still not sure about b and c.......