Induction is something like:Sai Alonzo said:Homework Statement
Prove the continuity from below theorem. I don't see a theorem. Do you ?
Homework Equations
As in the other thread: which relationships apply here ?
The Attempt at a Solution
So I've defined my {Bn} already and proven that it is a sequence of mutually exclusive events in script A. I need to prove that U Bi (i=1 to infinity) is equal to U Ai (i=1 to infinity) to use the Countable Additivity formula.
My prof states that I need to prove the 2nd property of Countable Additivity by Induction but I'm not really sure how to go about.
Sai Alonzo said:Homework Statement
Prove the continuity from below theorem. If {An} is a monotone nondecreasing sequence of events in A and lim An is in A (n-->infinity) then
the probability of the countable union of An = probability of the lim An, n--> infinity = lim P(An), n--> infinity
Homework Equations
Assumptions
A1 is a subset of A2 which is a subset of A3 and so on.. (monotone nondecreasing sequence)
{An} is in A
lim An is in A, n--> infinity
The Attempt at a Solution
So I've defined my {Bn} already and proven that it is a sequence of mutually exclusive events in script A.
I need to prove that U Bi (i=1 to infinity) is equal to U Ai (i=1 to infinity) to use the Countable Additivity formula.
My prof states that I need to prove the 2nd property of Countable Additivity by Induction but I'm not really sure how to go about.
BvU said:Well, I don't really feel qualified for the contents of the statements, but: do you recognize what I wrote in #2 about the method of induction ?
The Continuity From Below Theorem is a fundamental theorem in real analysis that states that if a function is continuous at every point from below a given point, then it is also continuous at that point.
The Continuity From Below Theorem is used to prove the continuity of functions at a specific point. It is also used in the construction and analysis of more complex mathematical concepts, such as limits, derivatives, and integrals.
The proof for the Continuity From Below Theorem involves using the definition of continuity and the properties of limits to show that if a function is continuous at every point below a given point, then it must also be continuous at that point.
A strong understanding of basic calculus and real analysis is necessary to fully comprehend the Continuity From Below Theorem and its proof. Familiarity with concepts such as limits, continuity, and the properties of functions is also important.
The Continuity From Below Theorem has many applications in fields such as physics, engineering, and economics. It is used to model and analyze real-world phenomena, such as the behavior of fluids, the flow of electricity, and stock market trends.