corrigé.

ARA
M x ∈A x ≤M A
Asup A
Asup A S
x∈A x ≤S
M x ∈A x ≤M S ≤M
A A
sup A= +∞sup ∅=−∞
ARSsup A=S
x∈A x ≤S
ε > 0x∈A S −ε<x
M A S
M=S−ε S
(an)n≥1p≥1sp= sup{an:
n≥p}
(sp)p≥1
[−∞,+∞]
(an)n≥1lim an
(an)n≥1lim an=−∞ lim an= +∞
lim an(an)n≥1
(bk)k≥1(an)n≥1
n1< n2< . . . k ≥1
bk=anklimk→∞ bk≤limn→∞ an
(an)n≥1lim an
({an:n≥p})p≥1
sup
A⊂B⊂Rsup B A sup A

[−∞,+∞]
lim an=−∞ sp→ −∞ A > 0
p sp≤ −A∀n≥p, an≤ −Alim an=−∞
lim an=−∞
lim an= +∞sp→+∞
sp= +∞p≥1an
lim an= +∞an
(an)n≥1c
lim an> c c0> c an> c0
n
lim an< c c0< c an< c0
n
(Xn)n≥1X= limn→∞ Xn
a, b a < b {ω∈Ω : a <
X < b}
lim an> c (an)n≥1
lim an(ank)k≥1c0c00 c <
c0< c00 <lim an(ank)k≥1lim ank
ank≥c00 > c0(an)n≥1
c0
c0> c n an> c0
(an)n≥1(bk)k≥1
c0
c0(bk)k≥1
(an)n≥1lim an≥lim bk≥c0> c
an< c0n an
c0an
c0lim an≤c0< c
c0< c
(an)n≥1c0n1≥1an1≥c−1
n2> n1an2≥c−1
2
(nk)k≥1k≥1ank≥c−1
k
(ank)k≥1
(an)n≥1
c c lim ank≥lim c−1
k=c
lim an≥lim ank≥c

a, b a < b
{a<X<b}={X > a}∩{X < b}
= [
k≥1
lim sup
n→∞ Xn> a +1
k!∩ [
k≥1
lim inf
n→∞ Xn< b −1
k!
= [
k≥1\
p≥1[ n≥pXn> a +1
k!∩ [
k≥1[
p≥1\ n≥pXn< b −1
k!.
{a<X<b}
(Xn)n≥11
Plim Xn
log n>1= 0
X1, X2, . . .
Plim Xn
log n<1= 0
lim Xn
log n
lim Xn0
lim Xn
log n>1=[
k≥1Xn
log n>1 + 1 k
=[
k≥1
lim sup
n→∞ Xn
log n>1 + 1 k.
n, k ≥1Xn1
PXn
log n>1 + 1 k=PXn>log n1+ 1
k=e−logn1+ 1
k=1
n1+ 1
k
.
k≥1n
X
n≥1
PXn
log n>1 + 1 k<+∞.
Plim sup
n→∞ Xn
log n>1 + 1 k= 0.

Plim Xn
log n>1= 0.
lim Xn
log n<1=[
k≥1Xn
log n<1−1 kn
=[
k≥1
lim inf
n→∞ Xn
log n<1−1 k
=[
k≥1lim sup
n→∞ Xn
log n≥1−1 kc
.
n, k ≥1
PXn
log n≥1−1 k=1
n1−1
k
,
X
n≥1
PXn
log n≥1−1 k= +∞.
X1, X2, . . .
Plim sup
n→∞ Xn
log n≥1−1 k= 1,
Plim Xn
log n<1= 0.
Xn=X1n≥1
Xn
log n=X1 log n0
lim Xn
log n= 0
P(lim Xn
log n<1) + P(lim Xn log n>1) + P(lim Xn log n= 1) = 1
lim Xn
n1
(an)
Xn< anPn≥1P(Xn< an)=+∞
a0P(X < a) = 1 −e−a=a+O(a2)
Pn≥1an= +∞an=1
n
{Xn≤1
n}1
lim Xn= 0 lim Xn= 0

Xn=X1n≥1 lim Xn=X1
l
l
X
X
(Xn)n≥1
X f :R→R
(f(Xn))n≥1f(X)
l
l
(an)n≥1l
ε > 0n0≥1n≥n0|an−l|> ε
(an)n≥1ε
l(bk)k≥1= (ank)k≥1k≥1
|bk−l|> ε β (bk)k≥1|β−l| ≥ ε
(bk)k≥1
l
(Xn)n≥1
X ε > 0P(|Xn−X|> ε) 0 n
α > 0 (nk)k≥1
k≥1P(|Xnk−X|> ε)> α (Xnk)k≥1(Xn)n≥1
X
(f(Xnk))k≥1(f(Xn))n≥1(Xnk)k≥1
X(Xnkl)l≥1
X f (f(Xnkl))l≥1
f(X) (Xnkl)l≥1X
(f(Xnkl))l≥1f(X)
(f(Xn))n≥1f(X)