# Limit Proofs (Sup and Inf)

• Sep 15th 2008, 02:53 PM
weakmath
Limit Proofs (Sup and Inf)
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)?
• Sep 15th 2008, 03:07 PM
icemanfan
For a), you need to define the sequence $A_n$ before you can draw any conclusions about it.

For b) and c), you need to define the sequences $B_n$ and $C_n$ before you can calculate them.
• Sep 19th 2008, 12:58 PM
weakmath
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.......