Math 554/703I - Analysis I
Test 1 - Solutions
1. List the axioms for the real numbers.
[See course lecture notes.]
2. Let Abe an nonempty subset of IR.
a.) Define ‘upper bound’ for A.
b.) Define ‘least upper bound’ for A.
c.) Prove that least upper bounds are unique.
[Suppose that α, β are both least upper bounds for A. Since αis an upper bound for A, and βis a
least upper bound of A, then β≤α. By symmetry, α≤β]
3. Prove for each a∈IR, (−a) = (−1) ·a.
[See course lecture notes: But use the fact that additive inverses are unique and observe that
a+ ((−1)(a)) = a(1 + (−1)) = a·0 = 0.]
4. Suppose that IR is an ordered field, prove
a) Multiplicative inverses are unique.
[If α, β are both a multiplicative inverse of an element a, then α=α1 = α(aβ) = (αa)β=β. ]
b) (ab)−1=a−1b−1
[Use the previous part, i.e. multiplicative inverses are unique, and observe that
(ab)(a−1b−1) = (aa−1)(bb−1) = 1. ]
c) If 0 < x < y, then x2< y2.
[Use 0 < x and multiply x < y to get x2< xy. Use 0 < y and multiply x < y to get xy < y2. Using
the transitive property, gives x2< y2.]
5. Negate the statement:
‘For each ǫ > 0 there exists a natural number Nsuch that for every pair x, y ∈[0,1]
which satisfies |x−y|<1/N , then |f(x)−f(y)|< ǫ.’
[There exists ǫ > 0 such that for each natural number Nthere exists a pair x, y ∈[0,1] which
satisfies |x−y|<1/N, but |f(x)−f(y)| ≥ ǫ.]
6. a) Prove that the natural numbers are not bounded.
[See course lecture notes.]
b) State and prove the Archimedean principle.
[See course lecture notes.]
c) Prove that for each ǫ > 0, there exists a natural number Nsuch that for all N≤n
there holds 1
n2< ǫ.
[Use the Archimedean Principle to find a natural number Nso that 1
N< ǫ. Notice that if n≥N,
then N≤n·1≤n·nand so 1
n2≤1
n≤1
N< ǫ]
7. Pick one: Sketch the proof that every open interval (a, b), where a < b, contains a rational
(irrational) number.
[See course lecture notes.]
8. a) State the triangle inequality for the real numbers.
[|a+b| ≤ |a|+|b|]
