Question

1) Prove that for all real numbers x and y, if x < y, then x < (x+y)/2 < y

2) Let a, b ∈ R. Prove that:

a) (Triangle inequality) |a + b| ≤ |a| + |b| (HINT: Use Exercise 2.1.12b and

Proposition 2.1.12, or a proof by cases.)

Answer #1

Prove the statement " For all real numbers r, if r is
irrational, then r/2 is irrational ". You may use any method you
wish. Be sure to state what method of proof you are using.

Let x and y be real numbers. Then prove that sqrt(x^2) = abs(x)
and abs(xy) = abs(x) * abs(y)

Prove the statement
For all real numbers x, if x − ⌊x⌋ < 1/2 then ⌊2x⌋ =
2⌊x⌋.

Real Analysis I
Prove the following exercises (show all your work)-
Exercise 1.1.1: Prove part (iii) of Proposition
1.1.8. That is, let F be an ordered field and x, y,z ∈ F. Prove If
x < 0 and y < z, then xy > xz.
Let F be an ordered field and x, y,z,w ∈ F. Then:
If x < 0 and y < z, then xy > xz.
Exercise 1.1.5: Let S be an ordered set. Let A
⊂...

Prove that |cos x - cosy| <
|x-y| for any x, y in the real numbers.

Define f: R (all positive real numbers) -> R (all positive
real numbers)
by f(x)= sqrt(x^3+2)
prove that f is bijective

Claim: If (sn) is any sequence of real numbers with
??+1 = ??2 + 3?? for
all n in N, then ?? ≥ 0 for all n in N.
Proof: Suppose (sn) is any sequence of real numbers
with ??+1 = ??2 + 3??
for all n in N. Let P(n) be the inequality statements ??
≥ 0.
Let k be in N and suppose P(k) is true: Suppose ?? ≥
0.
Note that ??+1 = ??2 +
3?? =...

****Please show me 2 cases for the proof, one is using
n=1, another one is n=2, otherwise, you answer will be thumbs
down****Hint: triangle inequality. Don't copy the online answer
because the question is a little bit different
use induction prove that for any n real numbers, |x1+...+xn|
<= |x1|+...+|xn|.
Case1: show me to use n=1 to prove it, because all the
online solutions are using n=2
Case2: show me to use n=2 to prove it as well.

Prove the following:
If x and y are real numbers and
x+y>20, then x>10 or y>10

Prove the following:
For any positive real numbers x and y, x+y ≥
√(xy)

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 5 minutes ago

asked 6 minutes ago

asked 10 minutes ago

asked 18 minutes ago

asked 18 minutes ago

asked 22 minutes ago

asked 23 minutes ago

asked 28 minutes ago

asked 28 minutes ago

asked 34 minutes ago

asked 36 minutes ago

asked 46 minutes ago