TD 1: Exercices sur les Nombres Réels - Analyse Mathématique

1
x, y ∈Rx≤y z ∈R.
||x|−|y|| ≤ |x±y|
x≤z≤y⇔ |x−z|+|z−y|+|x−y|.
2
x a |x−a|<|a|. a−|a|<
x<a+|a|x a.
x y |x| ≥ ||x−y|−|y||.
a, b c
a(1 −b), b(1 −c), c(1 −a)1
4.
3
∀x, y ∈R, E (x) + E(y)≤E(x+y)≤E(x) + E(y)+1
E(x) + E(−x)x∈R.
∀x, y ∈R, E (x) + E(y) + E(x+y)≤E(2x) + E(2y).
4
x∈R, N x < N ≤N+ 1.
N[x].
[x+n]x∈Rn∈N.
5
x̸=−1/3,
g(x) = 2x+ 1
3x+ 1.

x7→ g(x).
g(N) = {g(0) , g (1) , g (2) ,···}.
g(N)? g(Z)?
g(N)
g(Z)?
6
A⊂B⊂RB
A
sup(A)≤sup(B).
7
A=x∈R∗|x+1
x<2
sup A.
B=x∈R∗|x+1
x≤2
sup A.
8
A, B R.sup (A),
sup (B), , inf (A),inf (B),inf (A+B),
sup (A+B) = sup (A) + sup (B)
inf (A+B) = inf (A) + inf (B),
A⊂B⇒inf (B)≤inf (A) sup (A)≤sup (B)
|x−y|,(x, y)∈A2= sup A−inf A.
9
AR. A
A A.

A=1
n+ (−1)n, n ∈N∗sup Ainf A.
A=(−1)nn
n+ 1 n∈N.
10
Q R
r3, r ∈Q∈R.
a
10n, n ∈Na∈Z R.
11
R?R?
a)Zb)Qc) [0,1[ ]1,+∞[d)1
nn∈N∗e)1 nn∈N∗∪ {0}
12
R2
A=(x, y)∈R2;x > 0;B=(x, y)∈R2;xy = 1C=(x, y)∈R2;xy > 1
D=(x, y)∈R2;x2+y2≤2⧹(x, y)∈R2; (x−1)2+y2<1
13
A B R, A+B{a+b;a∈A b ∈B}.
A B A +B
R2
14
A=x∈R;∃p∈Z, q ∈Z, x =p+q√2
α=√2−1.

n∈Zx∈A, nx ∈A.
n∈N, αn∈A.
0< α < 1
2n∈N∗0< αn<1 n.
x, y ∈R.0< x < y. n ∈N
αn< y −x
a∈A x < a < y.
AR.