Théorème de Bolzano-Weierstrass

(In)n∈NIn= [an, bn]
In+1 ⊂In|bn−an|
Tn∈NIn
In
f(x)=0 f
[a, b]
[a, b]
f(a)f(b)≤0 [a, a+b
2] [a+b 2, b]
1
2{x0}
f(x0)=0

y=f(x)f
[a, b]y=c c
f(a)f(b)
fQ∩[0,2] Qf(x) = x2
f(0) = 0 f(2) = 4
Q
10−n
√2

ε
ARα
α A α
A A
A= [0,1[ 0 1 A[0,1]
0∈A1/∈A
A={1
n, n ∈N∗}0A0/∈A

A
[a, b]a<b [a, a+b
2] [a+b 2, b]
A
{x0}
x0A x0
A
α A R
A α
(un)n∈Nε > 0n0
]α−ε, α +ε[
α
α A α α
A
α A n ∈N∗]α−1
n, α+1 n[
A α unε > 0
n00<1
n0< ε n ≥n0
un∈]α−1
n, α +1 n[⊂]α−1 n0
, α +1
n0
[⊂]α−ε, α +ε[
(un)n∈N∗A α
n un6=α
AR
A