1. ## Continuity of P

Suppose P([0,∞)) = 1. Prove that there is some n such that P([0,n]) > 0.9
I cant figure this out.

2. ## Re: Continuity of P

Originally Posted by Kanwar245
Suppose P([0,∞)) = 1. Prove that there is some n such that P([0,n]) > 0.9
What is this limit $\lim _{n \to \infty } P\left( {\left[ {0,n} \right]} \right) = ?$.

3. ## Re: Continuity of P

Originally Posted by plato
what is this limit $\lim _{n \to \infty } p\left( {\left[ {0,n} \right]} \right) = ?$.
p([0,∞))

4. ## Re: Continuity of P

Originally Posted by Kanwar245
p([0,∞))
Do you understand the theory of limits?
This question is based on the definition of a limit.
In the definition use $L=1~\&~\epsilon=0.1$
If you don't know what that is all about, then you should not been asked to work this question.

5. ## Re: Continuity of P

Originally Posted by Plato
Do you understand the theory of limits?
This question is based on the definition of a limit.
In the definition use $L=1~\&~\epsilon=0.1$
If you don't know what that is all about, then you should not been asked to work this question.
i see i do know epsilon delta definition of a limit, but the way they proved it is with contradiction

6. ## Re: Continuity of P

Originally Posted by Kanwar245
i see i do know epsilon delta definition of a limit, but the way they proved it is with contradiction
You are given that $\lim _{n \to \infty } P\left( {\left[ {0,n} \right]} \right) = 1$.
That means that $\left( {\exists N} \right)\left[ {n \geqslant N\, \Rightarrow \,\left| {P\left( {\left[ {0,n} \right]} \right) - 1} \right| < 0.1} \right]$.

From there $1 - 0.1 < P\left( {\left[ {0,N} \right]} \right)$.