Raisonner/Rédiger

E
X
X X ∈E
fR(F)
x(♥)
f0(x) = −2x
(1+x2)2x f
]− ∞,0] [0,+∞[
f:R→R
x7→ 1
1+x2
(♥)f0(x)
x
F
X X ∈F

g:[0,+∞[→R
x7→ √x
g0(0)
g0
f
f0(x) (F)
X p(X) (A)
p(X)X
∀
∀X∈E p(X) = 4.(B)
(C)
∃X∈E:p(X)=5.(D)
A, B D X X
A, B, C D X C
X D
X
E
P
EP
x∈E
xP
EP
x x P
x E
x E
PE

n4k n = 4 ·k4 = 2 ·2
n= 2 ·(2 ·k) 2 ·k n 2
4
4
P Q
P Q P =⇒Q
P P Q
n k n = 2 ·k n2= 4k2= 2 ·(2k2)
2k2n2
n n n2
P=⇒Q
Q=⇒P
n n2
n k ∈Nn= 2k+ 1
n2= (2k+ 1)2= 2 ·(2k2+ 2k) + 1 (2k2+ 2k)n2
n n2n
P Q
P Q P ⇐⇒ Q
P⇐⇒ Q P ⇐⇒ Q P =⇒Q Q =⇒P

P P ⇐⇒ Q Q
P1⇐⇒ P2. . . ⇐⇒ Pnn
2k1n−1Pk⇐⇒ Pk+1
a b
|a+b| ≤ |a|+|b| ⇐⇒ p(a+b)2≤q(|a|+|b|)2
⇐⇒ (a+b)2≤(|a|+|b|)2
⇐⇒ a2+ 2ab +b2≤ |a|2+ 2|a|·|b|+|b|2
⇐⇒ ab ≤ |ab|.
ab |a+b| ≤ |a|+|b|
R
A B A ⊂B A
B x A
x A x B
n∈N∗UnnU
1
z∈Unz zn= 1 |zn|=|z|n= 1 |z| ∈ R+
x7→ xnR+R+xn= 1 1
R+|z|= 1 z∈U
Un⊂U
A=B A ⊂B B ⊂A
A=B A ⊂B B ⊂A
A⊂B∀x∈A x ∈A=⇒x∈B

P
X1
X2X1=X2
ARA A
M1M2A M1A
M2M1≤M2M2A
M1M2≤M1M1=M2
P P
P
P
√2
√2p q
√2 = p/q p q
2q2=p2p2
p p k p = 2k+ 1
p2= 4k2+ 2k+ 1 = 2(2k2+k)+1 p2
p r =p/2p= 2r2q2=p2= 4r2q2= 2r2
q2q p
p q
√2
√2
P
P P √2
p q