$30.00

## Description

Problem 1. We define the sequence (u_{n} )_{n}_{≥}_{0} the following way:

^{}

^{ } ^{u}_{0 } ^{= } ^{2}

_{1}

n

^{ } ^{u}^{n+1 } ^{= } ^{1} ^{+} _{1} _{+} _{u}

1. Compute u_{1} , u_{2 } and u_{3} .

^{for} ^{n} ^{≥} ^{0}

2. Prove that for every n ∈ Z such that n ≥ 0 we have 1 ≤ u_{n} ≤ 2.

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

_{1 } _{1 } _{1 } _{1 } _{1}

^{(1} ^{−} _{2} ^{)(1} ^{−} _{2}^{2} ^{)} ^{.} ^{.} ^{.} ^{(1} ^{−} _{2}^{n} ^{)} ^{≥} _{4} ^{+} _{2}^{n}^{+1} ^{.}

Problem 3. Let (u_{n} )_{n}_{≥}_{0} be the sequence of real numbers defined by

_{u}_{n} _{=} _{3} _{×} _{5}2n+1 _{+} _{2}3n+1

1. We are going to prove that

17 divides u_{n} for every n ∈ Z, n ≥ 0

by induction on n.

(a) For every n ∈ Z, n ≥ 0 compute u_{n+1} − 5^{2} · u_{n} and prove that it is a multiple of

17.

(b) Prove that 17 divides u_{n} 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 (u_{n} )_{n}_{∈}_{N } is

• bounded above when: ∃ M ∈ R such that (∀n ∈ N, u_{n} ≤ M ).

• bounded below when: ∃ m ∈ R such that (∀n ∈ N, u_{n} ≥ m).

• bounded when: (u_{n} )_{n}_{∈}_{N } 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 =⇒ |u_{n} − `| < ).

• converges towards +∞ when:

∀A > 0, ∃ m ∈ N such that ∀ n ∈ N, (n ≥ m =⇒ u_{n} > A).

1. Give an example of sequence of real numbers (u_{n} )_{n}_{∈}_{N } which is bounded above but not bounded below.

2. Let (u_{n} )_{n}_{∈}_{N } be a sequence of real numbers. Write in quantifiers the statement: (u_{n} )_{n}_{∈}_{N } is not bounded.

3. Write in quantifiers the statement:

(u_{n} )_{n}_{∈}_{N } does not converge towards +∞.

4. Show that the sequence of real numbers defined by

u_{n} = (−2)^{n } ∀ ∈ N

does not converge towards +∞.

5. Let (u_{n} )_{n}_{∈}_{N } be a sequence of real numbers. Write in quantifiers the statement: (u_{n} )_{n}_{∈}_{N } does not converge towards the real number `.

6. Show that the sequence of real numbers defined by

_{1}

^{u}^{n} ^{=} ^{1} ^{−} _{n}^{2 } ^{∀} ^{∈} ^{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.} ^{F}^{or} ^{e}^{v}^{ery} ^{∈} ^{R,} ^{>} ^{0,} ^{pr}^{ov}^{e} ^{that } ^{there} ^{is} ^{m} ^{∈} ^{N} ^{su}^{c}^{h} ^{that } _{m} ^{< } ^{.}

3. Prove that the sequence (u_{n} )_{n}_{∈}_{N } defined by

_{1}

^{u}^{n} ^{=} ^{√}_{n } ^{∀}^{n} ^{∈} ^{N}

converges towards 0.

See the definition of convergence in the previous problem.