1 Cours Suites de réels

E
A E
A E
R
R
R
R
H=Z+√2Z
e
un+1 =f(un)
f
f
un+1 = 1 + √un
un+1 = 1 −u2
n
u0< β
u0>−β
u0∈[0,1]

β < u0<−1
f(x) = x2−2
n= 2
x=n. ln x
(vn)
(un)
(un)
xn=x+n
(xn)

a b
ab ≤1
2a2+b2
(a+b)2≤2.a2+b2
√a+b≤√a+√b
max (a, b) = a+b
2+|a−b| 2
min (a, b)
|a| ≤ b a ≤b−a≤b
E
∀x, y ∈E, x ≤y y ≤x
A E
∀x∈A, x ≤M
A E
A A
A
A
A
A A
R
R
A= ]−∞,1[ B A
B= [1,+∞[B
A

A B A
B= [s, +∞[
ARs A
∀x∈A, x ≤s
∀ε > 0,∃x∈A∩]s−ε, s]
s−ε A
ARs A
s A
A s
X f X R
f f (X) sup f
f(X)
∀x∈X, f (x)≤Msup f≤M
Msupf
∀x∈R,arctan x < π
2
sup π
2
X f g X
R
sup (f+g)≤sup f+ sup g

sup f+ sup g f +g
∀x∈X, f (x)≤supf
∀x∈X, g (x)≤supg
∀x∈X, (f+g) (x)≤supf+ supg
M= supf+ supg f +g
sup (f+g)≤M= sup f+ sup g
X=R,f = cos2g= sin2
λ≥0 sup λf =λ. sup f
f
sup (−f) = −inf (f)
M ε > 0
n
M−ε≤un≤M
∀p≥n, M −ε≤up≤M≤M+ε
R
ERx y E
[x, y] = {(1 −t)x+ty/ t ∈[0,1]}
A E
∀x, y ∈A, [x, y]⊂A
R
f[a, b]Rf(a)>0f(b)<0
∃c∈]a, b[, f (c) = 0