Download Mathematical Proofs and Identities in R and more Assignments Mathematics in PDF only on Docsity!
Math 104 - Homework Problem Sets.
Adam Booth.
Summer 2007.
Week 1 – Axioms for the real numbers; sequences.
- Use mathematical induction to prove the following:
(a) 1 + 3 + 5 + · · · + (2n − 1) = n^2 ; (b) 3^4 n^ − 1 is divisble by 80; (c) n! > 2 n^ for n > 4. [Think: what should your base case be here?]
- Prove carefully from the axioms that the following hold for elements of R: [Two- column proofs or paragraph proofs are fine; you may quote any result which has been proved in class.]
(a) If ab = ac and a 6 = 0, then b = c; [This is referred to as the cancellability of multiplication.] (b) If ab = 0, then a = 0 or b = 0; [One nice way of doing this is by contradiction. This property of R is called being an integral domain.] (c) a ≥ b iff a + c ≥ b + c; (d) If a < b and c < 0, then ac > bc.
- This question is meant to help you appreciate an algebraic way in which R is special. (Much of the course is devoted to analytic ways in which R is special). Show that there does not exist a subset P ⊆ C satisfying the formal properties of positive numbers on page 11 of Burn.
- This question is about the absolute value function.
(a) Show that ||a| − |b|| ≤ |a| + |b|. Also state a ‘better bound’ on the left hand side [hint: look at the three results summarized on p. 21 of Burn]. Give examples of values for a and b which show that the left hand side, the better bound and the right hand side can all be same, or that some can be different. (b) Prove that max{a, b} = 12 (a + b) + 12 |a − b|; (c) Prove a similar identity for minimum.
- Some more practice with absolute values.
(a) Find all x ∈ R satisfying the following inequalities: i. 4 < |x + 2| + |x − 1 | < 5; ii. | 2 x + 1| ≤ |x − 1 |. (b) Find and sketch the set of all pairs (x, y) satisfying:
i. |x| − |y| ≥ 2; ii. 1 ≤ |xy| < 2.
- For each of the following sequences, say whether or not it’s monotonic (and if so, which of the four monotonicity properties it has) and whether it’s bounded, un- bounded or neither. If it’s not monotonic, give a monotonic subsequence.
(a) an = n. (b) bn = 4. (c) cn = (−1)nn (d) dn = b 600 /nc. [Note that, unlike Burn, I’m using ’floor notation’ – b·c – for the integer function, which he writes [·]. This is because I want to still be able to use square brackets as an alternative to (parentheses) when this helps make a formula easier to read, and because floor notation is more common and has the advantage of reminding you that we’re rounding down, not up.] (e) an = 100 − 20 n + n^2.
- Show that the following sequences are null. Do some using the definition (ie. by finding an N for each ) and some using the properties of null sequences listed on page 45.
(a) an = (^2) n^1 +. (b) bn = (^) n 2 n+. (c) cn = 2−n. (d) dn = n−^1 − n−^2.
- Adapt the method of question 3.40 from Burn to show that 2 n n! is null.
- Further adapt the method of question 3.40 to show that 2 n n^2 → ∞.
(a) Show that the union of two countable sets is countable. (b) By constructing a function from N onto N^2 (ie. the set of pairs of counting numbers), show that the union of countably many countable sets is countable. (c) Hence, show that the set of algebraic numbers (ie. those numbers which are the solution to some polynomial with integer coefficients) is countable. [You may assume that Zk^ is countable for any k.]
- Convergence of subsequences. Let (an) be some sequence and define dn := a 2 n+ (odd), en := a 2 n (even), tn := a 3 n (triples). Prove the following from the definitions, not using results about subsequences from class or the book.
(a) If an → l, then dn, en, tn → l. (b) If dn, en → l, then an → l. (c) If dn, en and tn are all convergent, then so is an. Give an example to show that it’s not enough just to have dn and en convergent.
- Determine whether the sets below are bounded above or below and, where possible, find their suprema and infima:
(a) { 2 n^ : n ∈ N}; (b) {(−1)n^ + (^1) n : n ∈ N}; (c) {sin(x) : x ∈ R}.
- Seeing how suprema interact with other concepts.
(a) Suppose A ⊆ B ⊆ R and B is bounded above. Prove carefully (assuming the appropriate version of the completeness principle) that A and B have suprema and that sup(A) ≤ sup(B). (b) Suppose, A, B ⊆ R are bounded above. Show that sup(A ∪ B) exists and satisfies sup(A ∪ B) = max{sup(A), sup(B)}. (c) Is it always true that sup(A ∩ B) = min{sup(A), sup(B)}?
- This questions tries to give a reason why taking convergence of decimals as our central completeness axiom may not have been the best idea. We have seen that a whole host of properties all hold in R and all fail in Q. What about it N? Decide which of the completeness principles (decimals, bounded monotonic subseqences, intersection of nested closed intervals, bounded sequences having convergent subsequences, cluster points, Cauchy sequences, l.u.b.s) hold in N.
Week 3 – Series.
Note: this problem set makes reference to some functions we haven’t formally defined yet. For these, you should state clearly what properties of the functions you’re using. Somewhere during this problem set, you may like to use that
n log(n) is divergent.^ [I can’t think of a way to prove this other than the integral test.]
- Evaluate the following series, or show that they’re divergent.
(a) ∑∞
n=
) 4 n+ .
(b) ∑ log
n + 1 n
(c) (^) ∑ sin(n).
- We saw last week that there are sequences which list all the rationals between 0 and 1 without repetition. Show that the sum of any such series is divergent.
- For which of the following sequences is
an convergent? [Hint: you might need to use that the sequence
( (^) n+ n
)n is convergent and converges to something greater than 1.]
(a) (^) n (^2) −n+1n+. (b) log( nn ). (c) log( n 2 n ). (d) (2 + (^1) n )−n. (e) (^) nnn!. (f)
√n+1−√n n.
- For which of the following sequences is
an convergent? Is it absolutely or condi- tionally convergent?
(a) (−1)n^ log( nn ). (b) (−1)t(n) 2 −n, where t is defined by
t(n) :=
0 n = 3m or n = 3m + 1 1 n = 3m + 2.
(c) sin( n 2 n ). (d) (−1)nbn where bn is given by b 1 = 10−^6 , bn+1 = sin(bn)/2.
Optional, hard. As in (d), but without the /2.
Optional, hard. 1 sin( nn ).
(^1) When I say hard, I mean it. I’ve thought about each of these for about five minutes, and can’t do them. If you can, let me know!
Week 4 – Continuity.
- Calculating limits.
(a) Find the limit of the following function as x → 1 from the left and the right. Do one of these using the sequences definition and one using the neighborhood definition. Does limx→ 1 f (x) exist? What relation does it have to f (1)?
f (x) :=
3 − x x > 1 1 x = 1 2 x x < 1.
(b) Use the algebra of limits to find limx→ 0 g(x), for the following g : R r { 0 } → R. [Note that the limit can be defined at a point not in the domain.]
g(x) :=
(1 + x)^2 − 1 x
(c) Recall,
χQ(x) :=
1 x ∈ Q 0 x /∈ Q.
Show that limx→ 0 f (x) does not exist, but limx→ 0 xf (x) does.
- Some preservation results.
(a) Prove that if f is continuous at x 0 , then |f | is too. (b) Is the reverse true? (c) Use part (a) together with a result from the first problem set to prove that if f and g are continuous functions, then so are max{f, g} and min{f, g}.
- Examples.
(a) Find a function f : R → R which is continuous everywhere except at the points of the set { (^) n^1 : n ∈ N} ∪ { 0 }. (b) Find a function g : R → R which is continuous everywhere except at the points of the set { (^) n^1 : n ∈ N}.
- Some applications of the IVT.
(a) Show that every polynomial of odd degree has a root in R. (b) Let f : [a, b] → R be continuous and suppose f (x) 6 = B for all x ∈ R. Show that it’s either the case that for all x, f (x) < B or for all x, f (x) > B.
- Some applications of the fact that the continuous image of a closed bounded interval is a closed bounded interval.
(a) Suppose f (x) → ∞ as x → ±∞ for some continuous f. Show that f has a global minimum. (b) Let f : [a, b] → R be continuous. Suppose that for each x ∈ [a, b], there is a y ∈ [a, b] such that |f (y)| ≤ 12 |f (x)|. Show that f has a root in [a, b].
- Suppose f : R → R is continuous and additive, ie. satisfies the identity
f (x + y) = f (x) + f (y).
Prove that there is some c ∈ R, such that f (x) = cx. Is the assumption of continuity necessary? [Hint (for first part): first show that f (0) = 0, f (−x) = x and use induction to show that f (nx) = nf (x) for n ∈ N. Then show that f (rx) = rf (x) for rational r. Deduce the result from this.]
- Suppose f : [a, b] → [c, d] is continuous, 1 : 1, and that f (a) = c, f (b) = d. Show that f must be strictly increasing.
- Use Taylor’s Theorem to obtain an approximation to
5 for which the error is at most 2−^9.
- Some more general applications of Taylor’s Theorem. In this question you may assume that f and g have as many continuous derivatives as you require.
(a) Suppose f : R → R satisfies |f (x)| ≤ 1 and |f ′′(x)| ≤ 1 on the interval [0, 2]. Show that |f ′(x)| ≤ 2 on this interval. [Hint: Consider the Taylor expansions of f (0) and f (2) about x ∈ [0, 2] with remainder involving f ′′.] (b) Suppose g′(0) = g′(2) = 0. Show that there is a c ∈ [0, 2] such that
|f ′′(c)| ≥ |f (2) − f (0)|.
[Hint: Note that |f (2) − f (0)| ≥ |f (2) − f (1)| + |f (1) − f (0)|. (Why?)]
Week 6 – Integration.
- (a) Suppose f (x) ≥ 0 for all x, f is continuous and
∫ (^) b a f^ = 0. Show that^ f^ (x) = 0 for all x ∈ [a, b]. (b) Show that the hypotheses of f (x) ≥ 0 and f continuous were required.
- Let f : [a, b] → R be a bounded function.
(a) Suppose f 2 is Riemann-integrable. Does this imply f is? (b) Suppose f 3 is Riemann-integrable. Does this imply f is? (c) Suppose we were considering
1 f^. Would your answer to (a) or (b) change?
- This question is about how to ‘adjoin’ certain types of functions to the class of Riemann integrable functions. If you’ve seen field theory in abstract algebra (note: this has nothing to do with the sorts of fields one encounters in multivariate calculus), this is rather parallel to the idea of adjoining a root of an equation there.
(a) Let R(χQ) denote the set of functions that can be written in the form
f (x) =
g(x) x ∈ Q h(x) x /∈ Q
for g, h Riemann integrable functions. For c ∈ [0, 1], define the c-integral, defined on all functions in R(χQQ) as
c
∫ (^) b
a
f := c
∫ (^) b
a
g + (1 − c)
∫ (^) b
a
h
where the integral on the left-hand side is the normal Riemann integral. Show that the c-integral is well defined and satisfies the conditions in Burn. (b) Given a function F whose range on every bounded set is bounded, show how to define a c-integral on R˙(F ), the set of functions which can be written as
f (x) = g(x) + λF (x)
where g is Riemann integrable and λ ∈ R.
- This question is about applying the techniques of integration theorems slightly more generally than you usually would in a calculus class.
(a) If f is a 1:1 continuously differentiable function with no zeroes, find a closed form expression for ∫ (^) a
b
f ′ f
(b) Suppose f and g both have infinitely many continuous derivatives and f (n)^ = f , g(m)^ = g (where n, m > 0 are the least such) and the least common multiple of n and m is odd. Find a closed form expression for ∫ f g.
Week 7 – Uniformity.
- Which of the following sequences of functions is converges uniformly on [0, 1]?
(a) xn (b) (x/2)n (c) sin(nx).
- (a) Prove that if fn : [a, b] → R are continuous, decreasing in n and fn → f which is also continuous on [a, b], then fn → f uniformly. (b) Is the same true for functions on the whole real line?
- Consider
fn(x) :=
0 x < (^) n+1^1 sin^2 (π/x) (^) n+1^1 ≤ x ≤ (^1) n 0 x > (^1) n
(a) Show that (fn) converges to a continuous function, but not uniformly. (b) By considering
fn, show that a series can converge absolutely on the whole real line without converging uniformly.
- Show that (^) ∞ ∑
n=
sin(nx) n^2 is continuous on R.
- Let (xn) be a sequence of distinct points from (0, 1) and
cn be an absolutely convergent series. Show that
cnχ(xn,∞)(x) is continuous at every x 6 = xn.
- Consider (^) ∞ ∑
n=
bnxc n^3
(a) Show that f is discontinuous on a countable dense set. (b) Show that f is Riemann-integrable on every bounded interval.
- Consider the following two initial value problems
y′^ = (1 − 2 x)y y(0) = 1 (1) y′^ = x^2 + y^2 y(0) = 0 (2)
(a) Find the first four terms in the Picard sequence of approximations for both problems. (b) For problem (1), show that the sequence converges uniformly for all x, and say what it converges to. (c) For problem (2), show that the sequence converges uniformly for |x| < 1 /
