Question

1) Prove: Any real number is an accumulation point of the set of rational number.

2) prove: if A ⊆ B and A,B are bounded then supA ≤ supB .

3) Give counterexample: For two sequences {a_{n}} and
{b_{n}}, if {a_{n}b_{n}} converges then
both sequences are convergent.

Answer #1

Prove Corollary 4.22: A set of real numbers E is closed and
bounded if and only if every infinite subset of E has a point of
accumulation that belongs to E.
Use Theorem 4.21: [Bolzano-Weierstrass Property] A set of real
numbers is closed and bounded if and only if every sequence of
points chosen from the set has a subsequence that converges to a
point that belongs to E.
Must use Theorem 4.21 to prove Corollary 4.22 and there should...

1. Prove that the sum of any rational number with an irrational
number must be irrational.
2. Prove or disprove: If a,b, and c are integers such that
a|(bc), then a|b or a|c.

True or False: Any finite set of real numbers is complete.
Either prove or provide a counterexample.

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.

1.- Prove the following:
a.- Apply the definition of convergent sequence, Ratio Test or
Squeeze Theorem to prove that a given sequence converges.
b.- Use the Divergence Criterion for Sub-sequences to prove that
a given sequence does not converge.
Subject: Real Analysis

Prove that between any two rational numbers there is an
irrational number.

10. (a) Prove by contradiction that the sum of an irrational
number and a rational number must be irrational. (b) Prove that if
x is irrational, then −x is irrational. (c) Disprove: The sum of
any two positive irrational numbers is irrational

Prove the following using the specified technique:
(a) Prove by contrapositive that for any two real numbers,x and
y,if x is rational and y is irrational then x+y is also
irrational.
(b) Prove by contradiction that for any positive two real
numbers,x and y,if x·y≥100 then either x≥10 or y≥10.
Please write nicely or type.

Prove if on the real number line R , set A = 0, B = 1, X = x and
Y = y (for some x , y ∈ R ) then the condition that X , Y are
harmonic conjugates with respect to A , B (i.e. ( A , B ; X , Y ) =
− 1) means 1/x + 1/y = 2

(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

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