1. The problem statement, all variables and given/known data "There is a very useful way of describing the points of the closed interval [a,b] (where we assume, as usual, that a < b) a. Consider the interval [0,b]. Prove that if x is in [0,b] then x = tb for some t with 0 <= t <= 1. What is the significance of the number t? What is the mid-point of the interval [0,b] b. Now prove that if x is in [a,b], then x = (1-t)a + tb for some t with 0 <= t <= 1. Hint: Can also be written as a + t(b-a). What is the midpoint of the interval [a,b]? What is the point 1/3 of the way from a to b? c. Prove, conversely, that if 0 <= t <= 1, then (1-t)a + tb is in [a,b] 2. Relevant equations N/A 3. The attempt at a solution I think I'm really close and I'm just missing something extremely stupid. I solved part a like so 0 <= x <= b 0/b <= x/b <= b/b 0 <= x/b <= 1 Let x/b = t 0 <= t <= 1 I'm stuck on part b however. No matter how many times I did the algebra I just never could come up with the right equation. I assume to start out like this: a <= x <= b a/b <= x/b <= b/b Let x/b = t a/b <= t <= 1 a <= tb <= b From here I'm somewhat stuck. I see that "tb" is part of the equation I"m looking for, I just need a (1-t)a from there. I also think I have solved c: 0 <= t <= 1 0 * (b-a) <= t * (b-a) <= 1 (b-a) 0 <= t(b-a) <= b -a a <= a + t(b-a) <= b Thanks for all of your help in advance!