MATH 4031 Final Short Answer

Find limn→∞ ∫[π/4,π/2] cosⁿ(x)dx

0

Let f(x) = sin x and g(x) = cos x on [0, π]. Find the scalar product <f,g>.

0

Let Fn(x)=1-xⁿ. Does the sequence Fn converge uniformly on [0,1)?

No. Fn→1 point wise on [0,1) but ‖Fn-1‖sup=1

Give an example of a sequence Xn with Xn>0 for all n ∈ N and Xn→0, yet Xn is not monotone.

Xn= [2+(-1)ⁿ]/n = ¹/₁, ³/₂, ¹/₃, ³/₄, ¹/₅,...

Find limn→∞ (n-1)/n

limn→∞ (n-1)/n = limn→∞ (1-1/n) = 1

Let f(x) = {1, x ∈ Q ∩ [a,b], -1, x ∈ [a,b]\Q. Find the numerical values of ∫[a,b]f (upper and lower)

upper = b-a lower = a-b

Give an example of an open cover O = {On | n ∈ N} of the interval (-1,1) that has no finite sub cover.

°{(-1+(1/n), 1-(1/n)) | n ∈ N}, °{(-1, 1-(1/n)) | n ∈ N} °{(-1+(1/n),∞) | n ∈ N}

Let ε > 0. How large must N ∈ N be to ensure that (1-1/n)/√n < ε for all n ≥ N?

Pick N > 1/ε². Such N exists since the set N is not bounded above.

Find the set of all cluster points of the set Q of all rational numbers.

R. This is true for any dense set.

Let T(f) = ∫[0,1] f(x)*(2/(1+x²))dx for each f ∈ R[0,1]. Find K ∈ R such that |T(f)| ≤ K‖f‖sup for all f ∈ R[0,1]. Hint: K arises naturally as a constant times π.

The constant K=2(arctan 1 - arctan 0) = π/2

Give an example of a decreasing nest of non-empty open intervals (a₁,b₁)⊇(a₂,b₂)⊇... such that bk-ak→0, yet ∩[k=1,∞) (ak,bk) ≠ 0.

(ak,bk)=(-1/k,1/k)

Give an example of an interval on which f(x)=x² is not uniformly continuous.

Any infinite interval will do.

True or give a counterexample: if f ∈ R[a,b], then f has only finitely many discontinuities in [a,b].

Counterexample with [a,b] = [0,1]: f(x) = {0 if x=0; 1/n if x ∈(1/(n+1), 1/n]. this function is integrable because it is monotone.

True or give a counterexample: If Xn increases and Yn increases, (Xn-Yn) is monotone.

Counterexample: Xn=2n +(-1)ⁿ =1,5,5,9,9,13,... Yn=2n -(-1)ⁿ =3,3,7,7,11,11,... Xn-Yn=2(-1)ⁿ =-2,2,-2,2,-2,2,...

True or give a counterexample: if |f| ∈ R[a,b], then f ∈ R[a,b]

Counterexample: f(x) = {1 if x=[a,b]∩Q; -1 if x∈[a,b]\Q}

True or Give a Counterexample: Xn + Yn converges if and only if both Xn and Yn converge.

Counterexample: If Xn = (-1)ⁿ, Yn= (-1)ⁿ⁺¹ then Xn+Yn → 0 but Xn and Yn diverge.

True or give a Counterexample: If xn >a for all n∈N and if xn →L as n→∞ then L>a.

Counterexample: Let xn = 1/n → 0 as n→∞, although xn > 0 for all n ∈ N.

True or Give a Counterexample: A bounded sequence times a convergent sequence must converge.

Counterexample: Xn = (-1)ⁿ is bounded and Yn ≡ 1 converges, but XnYn diverges.

True or give a counterexample: If Yn ≠ 0 for all n ∈ N and XnYn and Xn/Yn both converge, then the sequences Xn and Yn converge.

Counterexample: Xn = Yn = (-1)ⁿ

True or False: If x ∈ Q, then x√2 is irrational.

False, though it is true for x ≠ 0.

True or False: If E = {1/(n+1) | n ∈ N} then O = {(0,1/n) | n ∈ N} is an open cover of E that has no finite sub cover.

False. The single interval (0,1) ∈ O covers E.

True or False: The set of all polynomials of degree greater than or equal to 1 is a vector space.

False: Any polynomial minus itself has degree zero.

Suppose f(x)= {0 if x=0, (-1)ⁿn if x∈( 1/(n+1), 1/n], n ∈ N. True or False: f ∈ R[0,1], even though f has infinitely many discontinuities.

False: This function is unbounded and thus cannot be Riemann integrable.

True or False: the sequence 0,1, ½,0, ¹/₃, ²/₃,1, ¾, ½, ¼,0, ¹/₅, ²/⁵, ³/₅, ⁴/₅,1,... is a Cauchy sequence.

False: This sequence does not converge.

True or False: The sequence Fn(x) = 1 + xⁿ converges uniformly on [0, 1).

False: ‖Fn−1‖sup = 1 for all n ∈ N.

Give an example of a countable set S that is dense in R. Is your example closed, open, or neither?

For example, S = Q, which is neither open nor closed.

Give an example of a sequence of functions fn ∈ C[0,1] converging uniformly to 1 on [0,b] for all 0≤b<1 but not converging uniformly on the interval [0,1).

For example, fn(x) = 1-xⁿ for each n ∈ N.

Give an example of a monotone function f(x) for which lim x→0⁺ f(x) > f(0) > lim x→0⁻ f(x).

For example, let f(x) = {|x|/x if x ≠ 0, 0 if x = 0}

Give an example of an open cover O = {On | n ∈ N} of the set Z of all integers, such that O has no finite sub cover.

For example, one could let On = (-n, n) for all n ∈ N.

Give an example of two divergent sequences Xn and Yn such that Xn+Yn converges.

For instance, Xn=n, Yn=-n

Let f(x) = {1 if x=2; 0 if x∈[1,3]\{2}. Find a partition P of [1,3] such that U(f,P)-L(f,P) < ¹/₉

For instance, P={1,1.95,2.05,3}. In general, if X(i-1)<2<Xi, then U(f,P)=Xi-X(i-1) and L(f,P)=0.

Let N be the set of all functions f on [0,1] such that f(x)≤0 for all x ∈ [0,1]. Is N a vector space?

No. For example, if f(x)=-1 for all x ∈ [0,1] then f ∈ N but (-1)f ∉ N

Let ε > 0. Find a δ > 0 such that |x − 2| < δ and |y − 2| < δ implies that |x − y| < ε.

Since |x − y| ≤ |x − 2| + |y − 2|, δ = ε/2 or less works.

Give an example of f ∈ R[a,b] such that ‖f‖₂=0 yet f(x)≠0 on [a,b].

Take any function which is 0 except at finitely many points.

Find the set of all cluster points of the set Z of all integers.

The empty set. For any x ∈ R, let m be the greatest integer less than x and n the least integer greater than x, and set δ=min{x-m,n-x}>0. There is no integer k such that 0<|k-x|<δ, so x is not a cluster point in Z.

Let Fn(x) = {2n²x if 0≤x≤1/n 0 if 1/n<x≤1 for n > 1. Find a) limn→∞ Fn(x) for each x ∈ [0,1] b) limn→∞ ∫[0,1] Fn(x)dx

The first answer is 0 and the second answer is 1.

Let Fn(x) = {n if 0 < x < 1/n , 1 if x ∈ [0, 1]\(0, 1/n) } for all n ∈ N. For each x ∈ [0,1], find lim Fn(x) as n→∞. Is the convergence uniform on [0, 1]?

The point wise limit is 1 but the convergence is not uniform.

True or give a counterexample: if Xn+Yn and Xn-Yn both converge, the sequences Xn and Yn must converge.

True, because Xn=[(Xn+Yn)+(Xn-Yn)]/2 and Yn=[(Xn+Yn)-(Xn-Yn)]/2

True or False: If f⁺, f⁻ ∈ R[a,b] then |f| ∈ R[a,b].

True, since f⁺ + f⁻ = |f|

Let Xn ≠ 0 for all n ∈ N. True or false: |Xn|→0 if and only if 1/(|Xn|)→∞

True.

True or False: Every sequence of real numbers has a monotone subsequence.

True.

True or False: The function f(x) = 1/x is uniformly continuous on the interval [b, ∞), for every b>0.

True.

Let f, g ∈ C[a,b]. True or False: If f(a) > g(a) and if f(b) < g(b) then there exists c ∈ (a,b) such that f(c) = g(c).

True. Apply the intermediate value theorem to f - g.

Suppose f and g are in C[a,b] and suppose [f(a)−g(a)][f(b)−g(b)] ≤ 0. True or False: There exists c ∈ [a,b] such that f(c) = g(c).

True. Consider the function h(x) = f(x) - g(x) on [a,b].

True or False: If xn < ∞ for all n ∈ N and if limn→∞ xn =L, then L < ∞.

True. Every convergent sequence of real numbers is bounded, forcing L to be finite.

Let Fn(x)=1-(x/n) and f(x)=1. True or False: Fn → F pointwise, but not uniformly, on R.

True. Point wise convergence is obvious, but ‖Fn-F‖sup = ∞

True or give a counterexample: if f ∈ C[a,b] there exists ⁻x⁻∈ [a,b] such that ∫[a,b]f(x)dx = f(⁻x⁻)(b-a)

True. See Exercise 3.10

True or give a counterexample: if f ∈ R[a,b], then |f| ∈ R[a,b]

True. See Exercise 3.22

True or give a counterexample: if f ∈ R[a,b] and f(x)≥0 for all x ∈ [a,b], then ∫[a,b] f(x)dx ≥0

True. See Exercise 3.6

Let f(x) = {sin(1/x) if 0<x≤1; 0 if x=0} True or False: f has the intermediate value property on [0,1]

True. This is because f is continuous on any interval [a,1] with 0<a<1 and takes on all possible values for arbitrarily small x>0.

True or False: If An is a countable set for each n ∈ N, the set A=U[n=1,∞)An is countable.

True. We have An={am,n | m ∈ N and n ∈N}. We can arrange these elements in a sequence by writing the finite lists a₁,k, a₂,k-1, .... , ak,₁ in order of increasing k. This list may have repetitions, but we know that can be dealt with.

Let ε > 0. Find a value of δ > 0 such that |√x - √a| < ε for all x, a ∈ [0, ∞) such that |x-a|<δ

We can pick δ≤ε² since |√x - √a|²≤|x-a|.

The function T: C[0,1]→R defined by T(f)=∫[0,1]f(x)(1+x²)dx is a bounded linear function. Find a constant K such that |T(f)|≤K‖f‖sup for all f ∈ C[0,1]

We can take K = ⁴/₃. This is because |T(f)|≤∫[0,1]f(x)(1+x²)dx≤∫‖f‖sup(1+x²)dx = 4‖f‖sup/3

Use the Cauchy-Schwarz inequality to find a number K such that ∫[0,π/2]√(xsinxdx)≤K.

We can take K=π/(2√2). In R[0,π/2], we have ∫[0,π/2]√(xsinx)dx≤‖√x‖₂‖√(sinx)‖₂, ‖√x‖₂²=∫[0,π/2]xdx = π²/8 and ‖√(sinx)‖₂²=∫[0,π/2]sinxdx = 1

Let f(x) = { sin π/x if 0<x≤1, 0 if x=0} If ε>0, find a value of b ∈ (0, 1] such that U(f, P) - L(f, P) < ε/2, where P is any partition of [0,b]

You can use any δ<ε/4

a. Give an example of a subset of R that is neither open nor closed. b. Give an example of a subset of R that is both open and closed.

a. A=[a,b). It is not open because a ∈ A but there is no positive r with (a-r,a+r) ⊆ A. Its complement (-∞,a)∪[b,∞) is not open by the same argument applied to b. b. Either ∅ or R

Let fn(x) = 1/(x ln n) for all x ∈ [0,∞). a. True or False: fn converges to 0 uniformly on [1, ∞). b. Find limn→∞ ∫[1,n] 1/(x ln n) dx

a. True: fn converges to 0 uniformly on [1,∞) since ‖fn-0‖sup = 1/(ln n) b. limn→∞ ∫[1,n] 1/(x ln n) dx = limn→∞ 1 = 1

Let fn(x) = {n, 0<x≤1/n, 0, x∈[0,1]\(0,1/n]} a. Find the point wise limit of the sequence fn(x) for each x ∈ [0,1]. b. True or False: limn→∞ ∫[0,1] fn(x)dx = ∫[0,1]limn→∞ fn(x)dx

a. fn(x) →0 for each x ∈ [0,1] as n→∞. b. False: limn→∞ ∫[0,1] fn(x)dx = 1 ≠ ∫[0,1]limn→∞ fn(x)dx = 0.

Let Fn(x)={n²x if 0≤x≤1/n; 0 if 1/n≤x≤1} Find: a. limn→∞Fn(x) for each x ∈ [0,1] b. limn→∞∫[0,1]Fn(x)dx

a. limn→∞Fn(x)=0 We have Fn(0)=0 for all n, and Fn(x)=0 for n>1/x if x ∈(0,1]. b. limn→∞∫[0,1]Fn(x)dx= ½ In fact, ∫[0,1]Fn(x)dx= ½ for all n.

Let f(x) = { (x³-8)/(x−2if x≠2, C if x = 2} Find the value of c that makes f ∈ C(R).

c = 12

Let f(x) = {x² if x∈Q; 2-x if x∉Q} Find all values of x at which f is continuous.

f is continuous only at x=-2 and 1. Since there exists sequences of both rationals and irrationals converging to any x, f will be continuous at x iff x²=2-x

Give an example of f ∈ C(-1,1) for which the range of f is an infinite open interval.

f(x) = x/(1-x²) has range (-∞,∞)

Find lim sup xn and lim inf xn if xn = (-1)ⁿn + cosnπ/n

lim sup xn = ∞ lim inf xn = -∞

Find limn→∞ ∫[0,π/3]sinⁿxdx

limn→∞ ∫[0,π/3]sinⁿxdx=0 Over the interval [0, π/3], we have ‖sinⁿx‖sup = sinⁿ(π/3) = (√3/2)ⁿ→0 as n→∞. Thus sinⁿx→0 uniformly on [0,π/3].

Find limn→∞ ∫[1,2] x + 1/(1+x²)ⁿ dx.

limn→∞ ∫[1,2] x + 1/(1+x²)ⁿ dx =∫[1,2] x dx = ¾

Let Xn=1−(1/√n) Find the smallest value of n ∈ N for which |Xn−1|<100.

n = 1001

Find sup(S) and inf(S) for S = {(-1)ⁿ⁻¹(1-(1/n)) | n ∈ N}

sup(S) = 1 inf(S) = -1

Let Fn(x) = x/(x+n) on the interval I = [0, ∞). Find ‖Fn - 0‖sup on I.

‖Fn - 0‖sup = 1

Let fn(x) = nxe^(-nx²) for all x ∈ R. Find ‖fn‖sup.

‖fn‖sup = √(n/2e)

Express limn→∞ (2/n) ∑[k=1,n] sin(1 + 2k/n) in the form ∫[a,b]f(x)dx.

∫[1,3]sinx dx, because this is limn→∞ ∑[k=1,n]f(a+k((b-a)/n)*((b-a)/n) for a=1, b=3, and f(x)=sinx. ∫[0,2]sin(1+x)dx also works.

Express lim 2/n ∑ (k=1, n) (2k/n)³ as an integral. Include the lower and upper limits of integration.

∫² x³ dx = 4 but the evaluation is not required.