Download Homework 2 (Due Tuesday October 6)

HOMEWORK 2: DUE TUESDAY OCTOBER 6
INSTRUCTOR: PAUL APISA
1. Practice with Evaluating Limits of Sequences
Exercise 1: All odd problems 1-39 in Section 11.4.
Exercise 2: Problems 53 and 54 in Section 11.4.
Exercise 3: If r is a real number then let (an ) be the sequence with an = rn . When
does this sequence converge? When this sequence converges what does it converge
to? Prove all claims using the -δ definition of convergence.
2. Theoretical Questions
Exercise 4: Show that a bounded increasing sequence is convergent. (Hint: if (an ) is
the sequence, then supn {an } should be the limit. This problem is a great exercise
in using proof by contradiction).
Exercise 5: If (an ) and (bn ) are convergent sequences (converging to A and B respectively) then show that the following are convergent sequences:
(1) (an + bn )
(2) (an · bn )
(3) When B 6= 0, b1n .
Determine what each sequence converges to.
Exercise 6: Let f : R −→ R be a function. Show that f is continuous if and only if
for every convergent sequence of reals xn −→ x it follows that the sequence (f (xn ))
converges to f (x). (Hint: this problem requires using both the -δ definition of
continuity and the -N definition of convergence and playing them off each other).
3. (Optional) Practice with (hopefully very short) Proofs
These problems are entirely optional and may be submitted any time before the first
midterm. Each correct solution will be worth bonus points. The number of bonus points
will correspond to a third of the value of a graded theoretical question.
Exercise 6: Write a proof by induction to show that the sum of the first n integers
is n(n+1)
.
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Exercise 7: What goes wrong in the following proof:
Proof. We will show that all cows are the same color. Proceed by induction on the
claim that any n cows are the same color. For n = 1, which is the base case, this is
trivially true. Now suppose that the claim holds for any collection of n cows and we
will show that the claim holds for any collection of n + 1 cows. Let {c1 , . . . , cn , cn+1 }
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be a set of n + 1 cows. We want to show that all cows are the same color. By the
induction hypothesis, all cows {c1 , . . . , cn } are the same color (say brown for the
sake of concreteness). By the induction hypothesis, all cows {c2 , . . . , cn , cn+1 } are
the same color as well. Since this second set contains cn , which is a brown cow,
it follows that all the cows in {c2 , . . . , cn , cn+1 } are brown. Therefore, all cows in
{c1 , . . . , cn+1 } are the same color. Therefore, all cows are the same color.
Exercise 8: Show that among any four integers there are two whose difference is
divisible by three. (Hint: use the pigeonhole principle)
Exercise 9: Show that if p is a prime number and p divides n2 (where n is an integer),
then p divides n. (Hint: use unique prime factorization of any integer; this statement
was implicitly used in showing the irrationality of the square root of two).
Exercise 10: Show that there are infinitely many prime numbers (Hint: suppose not
to a contradiction, then multiply all the primes together and add one. Is this new
number prime?)
Exercise 11: Show that the sum of two bounded sequences is bounded. Is the sum of
two unbounded sequences unbounded? If yes, prove it; if no, find a counterexample.
Exercise 12: Show that a convergent sequence is bounded.
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