Parties de R

R
R
R
R Q
+× ≤
•+×
•+×a, b, c a + (b+c)=(a+b) + c a(bc)=(ab)c
•+×a, b a +b=b+a ab =ba
• × +a, b, c a(b+c) = ab +ac = (b+c)a
R Z
•R R
R
Q
≤R
+×
R

R
N N ={0,1,2, . . .}Z={0,1,2, . . .} ∪ {−1,−2, . . .}
N Z
N Z
x x bxc
x
bπc= 3 b−πc=−4
x
bxc ≤ x < bxc+ 1.
x∈R
x−1<bxc ≤ x.
Rx
ε > 0n∈Nnε > x

p
10kp∈Zk∈N
D
3/10
x∈Rn∈Ndn(x) := b10nxc
10n
dn(x)≤x<dn(x) + 10−n.
dn(x)dn(x) + 10−n
x10−n
π= 3,1415926... d3(π)
x ε > 0d
x−ε<d<x+ε.
D
∀a<b∈R D∩]a, b[6=∅.
D R R
R
•x∈Rε > 0n∈N10nε > 1n=bln(1/ε)
ln(10) c
n ε < 10−nd=dn(x)
d−ε<d≤x≤d+ 10−n< d +ε,
|x−d|< ε
x−ε<d<x+ε.
x ε
•a b a < b x =a+b
2ε=b−a 2
]a, b[=]x−ε, x+ε[

p
q
p∈Zq∈N∗Q R \Q
5 3/10 10/3 23/222 √2
Q ( R e π
D⊂Q.
Q R
√2
R
AR
∃M∈R∀x∈A x ≤M.
AR
∃m∈R∀x∈A x ≥m.
A A
1 [0,1] 2 π

AR
•Asup(A)A
•Ainf(A)A
R
R R
R R
ARα∈R
α= sup(A)⇐⇒ α A
∀ε > 0,∃x∈A:α−ε<x≤α
A= [0,1[ B={r∈R\Q:r < 2}sup(A) sup(B)
A={1/n −1/p, n, p ∈N∗}sup(A) inf(A)
R Q
A=r∈Q:r2<2Q
Q
M A R
sup(A)≤M,
M A
sup(A)Asup(A)≤M
sup(A) = sup(B)⇐⇒ sup(A)≤sup(B) et sup(B)≤sup(A).