corrigé du TD n 5 : Nombres réels

◦
f:x∈R+7→ x−sin(x)f x ∈R+f0(x) =
1−cos(x)≥0f x ∈R+
f(x)≥0 sin(x)≤x g :x∈R+7→ x+ sin(x)
x∈R+−sin(x)≤x
∀x∈R+,|sin(x)| ≤ x.
sin x∈R−|sin(x)|=|sin(−x)| ≤ |−x|=|x|
∀x∈R,|sin(x)|≤|x|.
x, y ∈Rsin(x)−sin(y)
sin(x)−sin(y) = 2 cos x+y
2sin x−y 2.
t∈R|cos(t)| ≤ 1
|sin(x)−sin(y)| ≤ 2 sin x−y
3
≤ |x−y|.
{n+ 2mπ, (n, m)∈N×Z}R
α= 1/2π{nα +m, (n, m)∈N×Z}Rx∈R
ε > 0 (n, m)∈N×Z
x
2π−n 2π−m
≤ε
2π.
2π
|x−n−2mπ| ≤ ε.
y∈[−1,1] ε > 0x∈Ry= sin(x) (n, m)∈N×Z
|x−ny −2mπ| ≤ ε
|sin(x)−sin(n+ 2mπ)|≤|x−n−2mπ| ≤ ε
|x−sin(n)| ≤ ε.
(sin(n))n∈N[−1,1]