MATH 295 PROBLEMS 2

IGOR KRIZ

1. Solve the equation

|x + 1| − |x| + 3|x − 1| − 2|x − 2| = x + 2.

2. Prove that for any real number r,

sup{q ∈ ℚ|q < r} = r.

Here ℚ denotes the set of all rational numbers. [Hint: the case r = \sqrt{2} was done in class.]

3. Let A and B be two nonempty sets of numbers which are bounded above, and let A + B denote the set of all numbers x + y with x in A and y in B. Prove that

sup(A + B) = sup(A) + sup(B).

[Hint: The inequality sup(A + B) ≤ sup(A) + sup(B) is easy. Why? To show the other inequality, it suffices to show that sup(A + B) ≥ sup(A) + sup(B) − ϵ for every ϵ > 0. Begin by choosing x ∈ A and y ∈ B such that x ≥ sup(A) − ϵ/2, y ≥ sup(B) − ϵ/2.]

4. Prove by induction that the following formula holds for all natural numbers:

1^{3} + 2^{3} + ... + n^{3} = (1 + 2 + ... + n)^{2}.

5. The Fibbonacci sequence a_{1}, a_{2}, a_{3}, ... is defined as follows: a_{1} = 1,

a_{2} = 1,
a_{n} = a_{n−1} + a_{n−2} for n ≥ 3.

Prove by induction that

a_{n} = \frac{(\frac{1+\sqrt{5}}{2})^{n} − (\frac{1−\sqrt{5}}{2})^{n}}{\sqrt{5}} .