Question

**Let A ={1-1/n | n is a natural number}**

**Prove that 0 is a lower bound and 1 is an upper
bound: ** Start by taking x in A. Then x = 1-1/n
for some natural number n. Starting from the fact that 0 <
1/n < 1 do some algebra and arithmetic to get to 0 < 1-1/n
<1.

**Prove that lub(A) = 1: ** Suppose that
r is another upper bound. Then wts that r<= 1.
Suppose not. Then r<1. So 1-r>0. Use Archimedes
to show that there is a natural number n such that 1/n<
1-r. Use this to see that r<1-1/n. Thus...

**Prove that glb(A) = 0:** Suppose that s is
another lower bound. Then wts that 0 <=s. This is
easy to show because 0 is an element of A.

Answer #1

Exercise1.2.1: Prove that if t > 0 (t∈R),
then there exists an n∈N such that 1/n^2 < t.
Exercise1.2.2: Prove that if t ≥ 0(t∈R), then
there exists an n∈N such that n−1≤ t < n.
Exercise1.2.8: Show that for any two real
numbers x and y such that x < y, there exists an irrational
number s such that x < s < y. Hint: Apply the density of Q to
x/(√2) and y/(√2).

1. (a) Let S be a nonempty set of real numbers that is bounded
above. Prove that if u and v are both least upper bounds of S, then
u = v.
(b) Let a > 0 be a real number. Deﬁne S := {1 − a n : n ∈ N}.
Prove that if epsilon > 0, then there is an element x ∈ S such
that x > 1−epsilon.

Use strong induction to prove that every natural number n ≥ 2
can be written as n = 2x + 3y, where x and y are integers greater
than or equal to 0. Show the induction step and hypothesis along
with any cases

prove: a natural number n is prime if and only if sigma(n) =
n+1

Prove the following statement:
Suppose that p is a prime number and n is a natural number. If
n|p then n = 1 or n = p.

In number theory, Wilson’s theorem states that a natural number
n > 1 is prime
if and only if (n − 1)! ≡ −1 (mod n).
(a) Check that 5 is a prime number using Wilson’s theorem.
(b) Let n and m be natural numbers such that m divides n. Prove the
following statement
“For any integer a, if a ≡ −1 (mod n), then a ≡ −1 (mod m).”
You may need this fact in doing (c).
(c) The...

Using induction, prove the following:
i.) If a > -1 and n is a natural number, then (1 + a)^n >=
1 + na
ii.) If a and b are natural numbers, then a + b and ab are also
natural

Using field axioms and order axioms prove the following
theorems
(i) The sets R (real numbers), P (positive numbers) and [1,
infinity) are all inductive
(ii) N (set of natural numbers) is inductive. In particular, 1
is a natural number
(iii) If n is a natural number, then n >= 1
(iv) (The induction principle). If M is a subset of N (set of
natural numbers) then M = N
The following definitions are given:
A subset S of R...

Prove by induction that if n is an odd natural number,
then 7n+1 is divisible by 8.

1. Let A ⊆ R and p ∈ R. We say that A is bounded away from p if
there is some c ∈ R+ such that |x − p| ≥ c for all x ∈ A. Prove
that A is bounded away from p if and only if p not equal to A and
the set n { 1 / |x−p| : x ∈ A} is bounded.
2. (a) Let n ∈ natural number(N) , and suppose that k...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 3 minutes ago

asked 11 minutes ago

asked 19 minutes ago

asked 25 minutes ago

asked 57 minutes ago

asked 57 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago

asked 3 hours ago

asked 3 hours ago