Feuille d`exercices 2

n∈Nk∈Nk≤n
n
k=n! k!(n−k)!
n+1
k+1=n k+1+n kn
k
k n
Q
a, b, c, d ∈Qa≥b c ≥d a +c≥b+d
a, b, c, d ∈Qa≥b≥0c≥d≥0ac ≥bd
a, b ∈Qa≥b≥0a2≥b2n∈Nan≥bn
x /∈Qy∈Q⇒x+y /∈Q
x /∈Qy∈Q\ {0} ⇒ xy /∈Q
3
√2/∈Q
√3/∈Q
√3 + √7/∈Q
(√3 + √7)(√3−√7)
a√a /∈Q
a∈N√a /∈Q
a, b ∈N√a+√b√a−√b

Qx, y, z ∈Q
∀n∈N∗, x ≤y≤x+z
n,
x=y
A={1
n|n∈N∗} ⊆ QA
A
Xsup(X)X
X X
Xmax(X)
Xinf(X)X
X X
Xmin(X)
A
AQ
R
{2−1
n|n∈N∗}
{2 + (−1)n
n|n∈N∗}
{(−1)n+1
n|n∈N∗}
3n
3n+ 2 |n∈N
{x∈R|x2+x+ 1 ≥0}
{x∈R|x2+x−1≤0}
{x∈Q|x2>9}
A⊆RM= sup A
M A
∀ > 0,∃x∈A|x>M−
E F RE⊆F
inf F≤inf E≤sup E≤sup F.

ER[inf E, sup E]
[a, b]RE⊆[a, b]
A B R
sup(A∪B) = max(sup A, sup B)
sup(A∩B) = min(sup A, sup B)
sup(A∩B)≤min(sup A, sup B)
A B inf(A∪B) inf(A∩B)
ARα∈R
sup{α+x|x∈A}=α+ sup A
sup{αx |x∈A}=αsup A α ≥0α < 0
A, B R
A+B={x∈R|x=a+b a ∈A b ∈B}.
A B sup(A+B) = sup A+ sup Binf(A+B)
A B
A⊆RAc=R\A
A Ac
sup AcM∈RA⊇]M, +∞[
A Ac
Q R
∀a < b ∈R,∃q∈Q|a < q < b.
α∈R R
A={x∈Q|x<α}.
B={x∈Q|x≤α}
R
∀a<b∈R,∃x∈R\Q|a < x < b.
D
D=na
10k|a∈Z, k ∈No.
D ( Q D R

R
X={x∈Q|x2+x−1≤0}.
X
P XP={x∈Q|P(x)≤0}XP
XP
P XP
XP