Chapitre 2 : Nombres réels.

R
A= [0,1], B = [−1,1[, C =1−1
n, n ∈N∗, D =(−1)n−1 n, n ∈N∗,
E=Q∩[0,+∞[, F =Q∩]0,+∞[, G =N
A B R
∀x∈A , ∀y∈B, x ≤y
a= sup(A)b= inf(B)a≤b
sup(A) = inf(B)⇐⇒ ∀ > 0,∃x∈A , ∃y∈B , y −x≤
f: [0,1] →[0,1] f
a[0,1] f(a) = a
A={x∈[0,1] |f(x)≤x}A a ≥0
f(a)A
f(a)≤a f(a)∈A
a b
ab
√2√2
a=√2√2
b ab
√2√2

x y
max {x, y}=x+y+|x−y|
2
x|x|2=x2
x y |xy|=|x||y|
∀x∈R,∀y∈R,|x+y|≤|x|+|y|
r≥0x∈R|x| ≤ r⇔ −r≤x≤r
∀x∈R,∀y∈R,||x|−|y|| ≤ |x−y|
E(x+y) = E(x) + E(y)x y
E(x+y)−E(x)−E(y)
x > 0f(x) = E(x) + E(1
x)
x k ≥0
ak
ak10−k≤x<ak10−k+ 10−k
k≥0xk= 10−kakx
D R
R−Q R
a b a < b n
a < a +√2
n< b
R−Q R
R
f:R→Rx y
f(x+y) = f(x) + f(y)
α≥0x f(x) = αx