Sous-groupes de R

R
a b a > 0b > a
a
b
∀ ∃
∀x≥0,∃n∈Nna ∈]x, x +b[.
[0,1] 0 ]0,1]
]0,1]
x∈]0,1] ]0,1] x/2
0
A⊂R+A6=∅Ainf A
(∀x∈A, x ≥inf A ,
∀y > inf A, ∃x∈A:x < y ,
inf ]0,1] = 0 inf Q+
∗= 0

G⊂RG
0∈G ,
∀x∈G, −x∈G ,
∀x∈G, ∀y∈G, x +y∈G ,
G x ∈G n ∈Zn.x ∈G
rZ:= {nr / n ∈Z}={0, r, −r, 2r, −2r, 3r, −3r, ..., nr, −nr, ...}
R
{0},N,Z,Q,R,[0,1],]−1,1[,R,3Z
inf G∩]0,∞[= inf{x∈G / x > 0}.
r≥0rZ
GR
G=rZr≥0
R
RG
G6={0}
r= inf G∩]0,∞[= inf{x∈G / x > 0}.
r > 0x∈G x > r x ≥2r
r∈G G =rZ
r= 0
∀ > 0,∀x≥0,∃g∈G g ∈]x, x +[.
GR
α∈R−Q
Z[α] = {n+mα / n ∈Z, m ∈Z}R
{ei2παm / m ∈Z} {z∈C/|z|= 1}
R R
R
R

x+y y +x
Z[α]Z[α]
R
a
=∀x≥0,
= ]x, x +b[
={n.a / n ∈N}
=∃n∈N/ n.a ∈]x, x +b[
∀x≥0,∃n∈Nn.a ∈]x, x +b[.
n∈Nna ∈]x, x+b[n
n
x < na < x +b

x
a<n<x a+b a.
n x/a
x≥0n x/a
n > x/a n −1n
x/a n −1≤x/a b > a b/a > 1n≤x/a + 1 <
x/a +b/a
x
a< n < x a+b a
x < na < x +b.
na ∈]x, x +b[
inf A A
A
Ainf A
A
∀x∈]0,1[ x≥0
y∈Ry > 0
z= max(1
2,y 2).
z∈]0,1/2] z∈]0,1[ z≤y/2< y
∀z > 0,∃y∈]0,1[ y < z
inf A= 0
∀x∈Q∗
+x≥0
y∈Ry > 0
n=1
y
n≥1/y ny ≥1
z=E(ny)
n+ 1 =ny n+ 1 ,
1z∈Q+
∗z≤ny/(n+ 1)< y
inf Q∗
+= 0
∀n∈N, n.a ∈G n ∈N
0.a = 0 ∈G
n∈Nn.a ∈G n a ∈G
G G G n.a +a∈G
(n+ 1)a∈G n + 1
G na ∈G−na =
(−n).a ∈G
∀n∈Z, na ∈G
{0},Z={0,−1,1,2,−2,3,−3, ...},Q={a/b / a ∈Z, b ∈N∗},R,3Z
{0}Q N ]−1,1[

0∈ {0}
x∈ {0}x= 0 −x= 0 ∈ {0}
x∈ {0}y∈ {0}x= 0 y= 0 x+y∈ {0}
{0}
0∈Q
x∈Qa∈Zb∈N∗x=a/b
−x=−a
b,−a∈Z, b ∈N∗,
−x∈Q
x∈Qy∈Qa∈Zb∈N∗x=a/b a0∈Z
b0∈N∗y=a0/b0
x+y=a
b+a0 b0=ab0+a0b bb0ab0+a0b∈Z, bb0∈N∗,
x+y∈Q
Q
3∈N−3/∈N N
3/4∈]−1,1[ 3/4 + 3/4 = 6/4 = 3/2/∈]−1,1[ ] −1,1[
G={0} ⇒ G∩]0,∞[= ∅inf G∩]0,∞[
G=Z⇒G∩]0,∞[= {1,2,3,4,5, ....}inf G∩]0,∞[= 1
G=Q⇒G∩]0,∞[= {a/b :a∈N, b ∈N∗}inf G∩]0,∞[= 0 2.b).
G=R⇒G∩]0,∞[=]0,∞[ inf G∩]0,∞[= 0
G=rZ⇒G∩]0,∞[= {r, 2r, 3r, 4r, ...}inf G∩]0,∞[= r
G
inf G∩]0,∞[= 0 >0
R
R
r≥0rZ={rn :n∈Z}
0 = r.0∈rZ
x∈rZn∈Zx=rn −x=r.(−n)−n∈Z
−x∈rZ
x∈rZy∈rZn∈Zx=rn m ∈Zy=rm
x+y=r(n+m)n+m∈Zx+y∈rZ
rZ
r G∩]0,∞[r > 0
r∈G
]0,1[ G]r, 2r[
y∈G y ∈]r, 2r[
G]0, r[
0, r, 2r
y > r G∩]0,∞[