Analysis

Question 1

What is De Morgan’s Law?

Answer 1

Let X, Y ⊂ Z. Then we have:
Z(X ∪ Y ) = (Z\X) ∩ (Z\Y ),
Z(X ∩ Y ) = (Z\X) ∪ (Z\Y ).

Question 2

What is the triangle inequality?

Answer 2

|z1 + z2| ≤ |z1| + |z2| for all z1, z2 ∈ C.

Question 3

What is the definition of a limit and convergent sequence?

Answer 3

A real sequence (xn)n∈N has the limit x* ∈ R if,
for every ǫ> 0, there exists an index N ∈ N such that
|xn − x*| < ǫ for all n ≥ N.

Question 4

What is the uniqueness of limits theorem? (Prove)

Answer 4

Every convergent sequence (zn)n∈N has

precisely one limit. (proof in notes)

Question 5

What is theorem 3.4? (prove)

Answer 5

Every convergent sequence is bounded. (proof in notes)

Question 6

What is the squeezing theorem? (prove)

Answer 6

If |xn| ≤ yn for all n ∈ N and yn → 0 as n → ∞, then also xn → 0 as n → ∞.

Question 7

What is theorem 3.6? (prove)

Answer 7

Let xn → 0 as n → ∞. Let (yn) be a bounded sequence. Then we have xnyn → 0 as n → ∞

Question 8

Prove COLT

Answer 8

See notes

Question 9

What is the definition of the euler number?

Answer 9

e = lim( 1 + 1/n)^n

Question 10

What is theorem 3.9?(prove)

Answer 10

If xn → x* as n → ∞ and if f(x) is continuous at x*

, then we have f(xn) → f(x*) as n → ∞

Question 11

What is the generalised squeezing theorem? (prove)

Answer 11

Let (an),(bn),(cn) be three real sequences with an ≤ bn ≤ cn and let (an) and (cn) be convergent with the same limit, in other words, limn→∞ an = limn→∞ cn = x* Then (bn) is also convergent and we have limn→∞ bn = x*.

Question 12

Prove that if xn>=0 , lim(xn)>=0

Answer 12

Notes

Question 13

How do you negate a statement?

Answer 13

chnage each for every to a there exists and change the outcome to the opposite of itself.

Question 14

What is the contrapositive of if A then B?

Answer 14

if (notB) then (notA)

Question 15

What is the completeness axiom for the real numbers?

Answer 15

Every non empty subset of R that is bounded above has a supremum and every non empty subset of R that is bounded below has an infimum.

Question 16

What is theorem 6.2? (prove)

Answer 16

Let (xn) be a monotone increasing real sequence. If (xn) is bounded, then (xn) is convergent and we have
limn→∞xn = sup(X), where X = {xn | n ∈ N}

Question 17

What is proposition 6.4? (prove)

Answer 17

Let (xn) be convergent with limit limn→∞ xn = x* and (xnj) be asubsequence. Then (xnj) is also convergent and
limj→∞xnj = x*

Question 18

What is lemma 6.5? (prove)

Answer 18

Every real sequence contains a subsequence that is monotone increasing or a subsequence that is monotone decreasing.

Question 19

What is the Bolzano-Weierstrauss Theorem?

Answer 19

Let xn be bounded, then xn ha a subsequence that is convergent.

Question 20

What is a cauchy sequence?

Answer 20

For every epsilon greater than 0 there exists an N in the natural numbers such that for every n,m>N |xn-xm| is less than epsilon.

Question 21

What is theorem 6.8? (prove)

Answer 21

Let xn be cauchy. xn is then bounded.

Question 22

What is theorem 6.9? (prove)

Answer 22

Let xn be a convergent sequence. Then xn is cauchy.

Question 23

What is theorem 6.10? (prove)

Answer 23

Let xn be cauchy. Then xn is convergent.

Question 24

What is the definition of continuity of f(x) at c?

Answer 24

∀ E> 0 ∃ δ > 0 : |f(x) − f(c)| < E ∀ x ∈ X with |x − c| < δ.

Question 25

What is the preimage of a function?

Answer 25

f-1(Y0) = {x ∈ X | f(x) ∈ Y0}.

Question 26

What is proposition 7.5? (prove)

Answer 26

Let X ⊂ R, f : X → R and limx→c f(x) = A. Let (xn) be a sequence in X (in other words, xn ∈ X for all n ∈ N) with limn→∞ xn = c and xn <> c for all n ∈ N. Then we have limn→∞f(xn) = A.

Question 27

What is the definition for the limit of a function?

Answer 27

let X ⊂ R and f : X → R. For c ∈ R, we say that ”f(x) → A as x → c” or ”limx→c f(x) = A”, if the following holds: limx→c f(x) (i) For every δ > 0 the intersection ((c − δ, c) ∪ (c, c + δ)) ∩ X is not empty. (ii) For every > 0 there exists δ > 0 such that |f(x) − A| < ∀ x ∈ ((c − δ, c) ∪ (c, c + δ)) ∩ X.

Question 28

What is proposition 7.7? (Prove)

Answer 28

sup(f) + inf(g) ≤ sup(f + g) ≤ sup(f) + sup(g)

Question 29

What is the intermediate value theorem? (prove)

Answer 29

If f : [a, b] → R is continuous and f(a) < d < f(b), then there exists c ∈ [a, b] with f(c) = d. Likewise, if f(a) > d > f(b), then there exists c ∈ [a, b] with f(c) = d

Question 30

What is theorem 7.13? (prove)

Answer 30

Let f:[a,b] →R be continuous. f is bounded.

Question 31

What is theorem 7.14? (prove)

Answer 31

If f : [a, b] → R is continuous, then sup(f) exists and there exists c ∈ [a, b] with f(c) = sup(f).

Question 32

What is the limit superior of a sequence?

Answer 32

lim sup n→∞ (xn) = limn→∞(sup m≥n (xm)) = infn≥0(sup m≥n(xm)

Question 33

What is the definition of differentiable?

Answer 33

f(x) is differentiable at x=c if the limit as x tends to c of f(x)-f(c) /x-c exists.

Question 34

What is theorem 8.2? (prove)

Answer 34

if f(x) is differentiable then f(x) is continuous.

Question 35

What is Rolle's Theorem? (prove)

Answer 35

Let f : [a, b] → R be continuous and differentiable on (a, b) and suppose that f(a) = f(b). Then there exists c ∈ (a, b) such that f′(c) = 0

Question 36

What is the mean value theorem? (prove)

Answer 36

Let f be continuous on [a,b] and differentiable on (a,b). Then there exists a c in (a,b) s.t f'(c) = f(b) - f(a) / b-a

Question 37

Prove L'Hôpital's theorem (prove)

Answer 37

notes

Question 38

What is the mean value theorem for second order derivatives? (prove)

Answer 38

f cont on [a,b] , twice diff on (a,b), then there exists a c in (a,b) st f'(c) = 2((f(b)-f(a) -(b-a)f'(a))/(b-a)^2)

Question 39

What is lemma 9.2? (prove)

Answer 39

if the series of ak converges then lim(ak)=0

Question 40

What is the comparison test? (prove)

Answer 40

Let 0

Question 41

What is the absolute convergence theorem? (prove)

Answer 41

if the series of |ak| converges then the series of ak converges.

Question 42

What is the alternative sign test? (prove)

Answer 42

Let ak be a monotone decreasing sequence of numbers. Then if ak tends to 0, the series of (-1)^k+1 *ak is convergent.

Question 43

What is theorem 9.8? (prove)

Answer 43

The series of ak is convergent if Fk is convergent (integral test)

Question 44

What is the ratio test? (prove)

Answer 44

If |ak+1|/|ak| tends to a value less than 1 then the series of is convergent.

Question 45

What is the nth root theorem? (prove)

Answer 45

Let (ak) be a sequence with |ak|^1/k, tending to a as k tends to infinity. Then if a<1 the series of ak converges absolutely, else it diverges.

Question 46

What is the riemann rearrangement theorem?

Answer 46

If the series of ak is conditionally convergent, then there exists a rearrangement where the sum converges to c, and the sum can be rearranged to be divergent.

Question 47

What is theorem 9.12

Answer 47

the series of ak rearranged is equal to the series of ak if ak is absolutely convergent.