MATH 4650-5650 (Intro to Real Analysis) Fall 2023
HW 4, Due Sept 21, 2023
NAME:
1. (3 min’s) If u > 0 is any real number and x < y, show that there exists
a rational number r such that x < ru < y.
2. (5 min’s) Use the completeness property of R:
(MATH 5650 only) Show that t3 = 3 has a solution.
(MATH 4650 only) Show that t2 = 5 has a solution.
3. (5 min’s) Show that max{a + b, c + d} ≤ max{a, d} + max{b, c}. Is it
true that max{a+b+e, c+d+f } ≤ max{a, d}+max{b, c}+max{e, f }?
If yes, prove it.
1
4. Bonus Problem. A function f on a convex subset S ⊂ R is called
convex if
f (θx + (1 − θ)y) ≤ θf (x) + (1 − θ)f (y), ∀x, y ∈ S, 0 ≤ θ ≤ 1.
(a) Show that − log x is convex.
(b) Show that the generalized geometric-arithmetic inequality holds:
n
Πni=1 aαi i ≤
1X
ai ,
n i=1
αi , ai ≥ 0,
1 ≤ i ≤ n,
n
X
i=1
2
αi = 1.