Real Analysis

lim x->0+ sin(1/x) does not exist

Prove that limx→∞sin(x) D.N.E

Define A + B = {a + b|a ∈ A, b ∈ B}. Show that sup(A + B) = sup A + sup B

Show that if ∃a, ∀ε > 0; 0 ≤ a < ε then a = 0

Show that ∀ε > 0, ∃a ∈ A; a + ε> sup A

Suppose we have ∀a ∈ A, ∀b ∈ B; a < b. Show that sup A ≤ sup B.

Show that for a, b ∈ ℤ+ such that a < b, there exists unique x, y ∈ ℤ+ such that b = xa + y with 0 ≤ y < a

Show that for given a, b ∈ ℝ with b > a, there exists n ∈ ℤ such that na > b

Show that for every a ∈ ℝ there is n ∈ ℤ such that n > a

Show that ℤ is unbounded

Which of the following sets have the completeness axiom property ℤ,ℚ,ℝ∖ℚ

Prove that sup(a, b) = b and inf(a, b) = a.

|a + b| ^2 + |a − b|^ 2 = 2|a| ^2 + 2|b| ^2

||a| − |b|| ≤ |a − b|

|a| − |b| ≤ |a + b| ≤ |a| + |b|

|ab| = |a||b|

a^2 ≥ 0

See if | defines an order in ℤ