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Problem 1. We define the sequence (un )n0 the following way:

u0 = 2

1

n

un+1 = 1 + 1 + u

1. Compute u1 , u2 and u3 .

for n 0

2. Prove that for every n Z such that n 0 we have 1 un 2.

Problem 2. Chapter 10 Problem 22: for every n N, prove that

1 1 1 1 1

(1 2 )(1 22 ) . . . (1 2n ) 4 + 2n+1 .

Problem 3. Let (un )n0 be the sequence of real numbers defined by

un = 3 × 52n+1 + 23n+1

1. We are going to prove that

17 divides un for every n Z, n 0

by induction on n.

(a) For every n Z, n 0 compute un+1 52 · un and prove that it is a multiple of

17.

(b) Prove that 17 divides un for every n Z, n 0 by induction on n.

2. (not to be handed in) You can prove the same result by direct proof without induction

(using congruences).

Problem 4. 1. Prove that for every n Z, there exist a, b Z such that n = 5a + 2b.

2. Prove that for every n Z, there exist c, d Z such that n = 5c + 3d.

3. Prove, for every integer n 4, the following statement:

P (n): there exist nonnegative integers a, b such that n = 5a + 2b.

Hint: We suggest two methods. Explore both methods, choose your favorite one and write it up in your homework paper. In any case, you should be able to understand both solutions eventually.

Split into cases, depending on whether n is even or odd (then use induction).

Use double induction, which is a generalization of the standard induction.

The outline here is the following:

Base Step: check that the statement is true for n = 4 and n = 5.

Induction Step: prove, for every n 4, that (P (n) P (n + 1) = P (n + 2).)

4. Using similar methods as in question 3., prove for every integer n 8, the following statement:

Q(n): there exist nonnegative integers a, b such that n = 5c + 3d.

Problem 5. We say that a sequence of real numbers (un )nN is

bounded above when: M R such that (n N, un M ).

bounded below when: m R such that (n N, un m).

bounded when: (un )nN is bounded above and bounded below. We say that it

converges towards the real number ` when:

> 0, m N such that n N, (n m = |un `| < ).

converges towards + when:

A > 0, m N such that n N, (n m = un > A).

1. Give an example of sequence of real numbers (un )nN which is bounded above but not bounded below.

2. Let (un )nN be a sequence of real numbers. Write in quantifiers the statement: (un )nN is not bounded.

3. Write in quantifiers the statement:

(un )nN does not converge towards +.

4. Show that the sequence of real numbers defined by

un = (2)n N

does not converge towards +.

5. Let (un )nN be a sequence of real numbers. Write in quantifiers the statement: (un )nN does not converge towards the real number `.

6. Show that the sequence of real numbers defined by

1

un = 1 n2 N

does not converge towards 0.

Problem 6. We admit that for any real number x R, there exists a unique integer bxc Z

such that

1. What is b−2.6789c? bπc? b0c?

x [bxc, bxc + 1).

1

2. For every R, > 0, prove that there is m N such that m < .

3. Prove that the sequence (un )nN defined by

1

un = n n N

converges towards 0.

See the definition of convergence in the previous problem.