Cor1 - Jean-Romain HEU

fR
f
∀A∈R,∃x∈R,|f(x)|> A.
∃A∈R,∀x∈R,|f(x)| ≤ A
fR
∃a∈R,∃b∈R,∀x∈R\[a, b], f(x)≥0.
∃a∈R,∃b∈R,∀x∈R, x /∈[a, b] =⇒f(x)≥0.
x∈Rg(x) = x2−3h(x) = cos(x)
g A ∈Rx=p|A|+ 4 g(x) = |A|+1 > A
∀A∈R,∃x∈R,|g(x)|> A.
g a =−√3b=√3x∈R
x /∈[a, b]|x|>√3x2−3≥0g
[a, b]∃a∈R,∃b∈R,∀x∈R, x /∈[a, b] =⇒g(x)≥0.
h A = 1 ∀x∈R,|h(x)| ≤ A
h
a b
b≥a h [a, b]
π6= 0 n∈Nnπ > b nπ /∈[a, b]h(nπ) = cos(nπ) = −1
∀a∈R,∀b∈R,∃x∈R\[a, b], f(x)<0
∀k∈Z,∃j∈Z, k2+k= 2j.
k∈Zk2+k=k(k+ 1)
k k k + 1
k2+k
k

P
∀n∈Z,8n2−1 =⇒n .
R C P
R:∀n∈Z, n =⇒8n2−1.
C:∀n∈Z, n =⇒8n2−1.
P
P C
n∈Zn8|n2−1
n k n = 2k+ 1
n2−1 = (2k+ 1)2−1=4k2+ 4k= 4(k2+k)
k2+k j
k2+k= 2j
n2−1 = 8j8|n2−1
n8|n2−1
n2−1n
R
R n n2
n2−1n2−1
R
(un)n∈N
∀n∈N, un∈N
∀n∈N, un+1 ≥un
∃N∈N, uN< N
∃n∈N, un=n
n un< n
(un)n∈N
E={n∈N|un< n}E
NEN
n0
un0−1=n0−1
n0∈E un0< n0
n0E n0−1/∈E n0
u0≥0
un0−1≥n0−1
un0≥un0−1
n0−1≤un0−1≤un0< n0
n0−1n0
un0=un0−1=n0−1n
un=n

∀n≥0, un=1
2
u1<1
∀n∈N, un6=n
u0= 1
u1= 0 <1
∀n∈N, un6=n
∀n∈N, un=n+ 1 N
∀n∈N, un> n
x∈R+y∈R+2x+ 2y= 1 xy = 1
y= 1/2−x x(1/2−x)=1
x2−x/2 + 1 = 0 (x−1/4)2=−15/16 <0