Proof: Quantile Function Property

[icestone111]

Homework Statement

F^-1 is the quantile function of a general random variable X and has the following property that is analogous to the property of the c.d.f.
Prove: Let x₀ = lim_p→0,p>0 F^-1(p) and x₁ = lim_p→1,p<1 F^-1(p). Then x₀ equals the greatest lower bound on the set of numbers c such that Pr(X≤c) > 0, and x₁ equals the least upper bound on the set of numbers d such that Pr(X≥d) > 0.

The Attempt at a Solution

I'm not too sure where to begin with proving this problem...
My interpretation of the problem is that x₀ and x₁ are the (lowest?) lower and (greatest?) upper bounds such that Pr(X=x) > 0.

 

[lippyka]

Since F-1 is the quantile function of a general random variable X, this means that F-1(p) is the x-value such that Pr(X≤x) = p. Therefore, a limiting value of p→0 would result in an x-value such that Pr(X≤x) = 0 and a limiting value of p→1 would result in an x-value such that Pr(X≥x) = 0. Thus, x0 and x1 are the greatest lower bound and least upper bound respectively. Is this correct? If so, how would I go about proving it?