Question

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.

Answer #1

Prove that for all real numbers x, if x 2 is irrational, then x
is irrational.

Prove or disprove the following statements. Remember to disprove
a statement you have to show that the statement is false.
Equivalently, you can prove that the negation of the statement is
true. Clearly state it, if a statement is True or False. In your
proof, you can use ”obvious facts” and simple theorems that we have
proved previously in lecture.
(a) For all real numbers x and y, “if x and y are irrational,
then x+y is irrational”.
(b) For...

(1) Let x be a rational number and y be an irrational. Prove
that 2(y-x) is irrational
a) Briefly explain which proof method may be most appropriate to
prove this statement. For example either contradiction,
contraposition or direct proof
b) State how to start the proof and then complete the proof

In the style of the proof that square root of 2 is irrational,
prove that the square root of 3 is irrational. Remember, we used a
proof by contradiction. You may use the result of Part 1 as a
"Lemma" in your proof.

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.)

Use proof by contradiction to prove the statement given. If a
and b are real numbers and 1 < a < b, then
a-1>b-1.

Irrational Numbers
(a) Prove that for every rational number µ > 0, there exists
an irrational number λ > 0 satisfying λ < µ.
(b) Prove that between every two distinct rational numbers there
is at least one irrational number. (Hint: You may find (a)
useful)

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

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

Let E⊆R (R: The set of all real numbers)
Prove that E is sequentially compact if and only if E is
compact

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