Proofs for test 1

Question 1

Prove the following :

(f) ∀a; b ∈ R; (−a)(−b) = ab.

Answer 1

Look in book

Question 2

Prove the following :

(c) ∀a; b ∈ R; ab = 0 ⇔ a = 0 or b = 0.

Answer 2

Look in book

Question 3

Prove:
Proposition 1.2 Let S be a nonempty subset of R. If maximum or minimum of S exist,
then they are unique.

Answer 3

Look in book

Question 4

Prove the following :
Proposition 1.3 Let S be a nonempty subset of R.
1. If max S exists, then sup S exists, and sup S = max S.
2. If min S exists, then inf S exists, and inf S = min S.

Answer 4

Look in book

Question 5

Prove the following:

(a) lim C = C
(n->∞)

Answer 5

Look in book

Question 6

Prove that
(b) lim = lim an + lim bn = L +M.
(n->∞) (n->∞) + (n->∞)

Answer 6

Look in book

Question 7

Prove that
(c) lim(can) = c lim an = cL.
n→∞ n→∞

Answer 7

Look in book

Question 8

Prove that:
(d) lim (anbn) = (lim an)(lim bn) = LM
n→∞ (n→∞ )(n→∞)

Answer 8

LOOK IN THE BOOK

Question 9

Prove that if L ≠ 0, M = 0, Then
(lim an/bn) does not exist
(n→∞ )

Answer 9

Check the book brother

Question 10

State and prove the sandwich theorem

Answer 10

Check the book brother

Question 11

Prove that

(b) ∀a ∈ R; a · 0 = 0.

Answer 11

Check the book brother

Question 12

Prove that
(e) ∀a ∈ R; (−1)a = −a.

Answer 12

Check the book brother

Question 13

Prove that:
(d) ∀a; b ∈ R; (−a)b = −(ab).

Answer 13

Check the book brother

Question 14

Prove that:
(a) ∀a; b; c ∈ R; (a + b)c = ac + bc.

Answer 14

Check the book brother

Question 15

Prove that if M ≠ 0, lim (an/bn) =
n->∞

lim(an)/lim(bn)
n->∞ n->∞

Question 16

Prove that if an is a sequence with an>0 for all indices then

lim(an) = lim(1/an) = 0
n->∞ n->∞