2016-2017 Feuille 1 : Logique, ensembles et récurrence. 1) Parmi

BCP ST 1B
∀x∈[0,1] x2<1
∀k∈N∗k2>k+ 1
∀(a, b)∈(R+)2,√a+b6√a+√b
f[0; 1]
f[0; 1]
[0,1]
f[0; 1]
N
R
R
∀x∈R, a +bcos2(x) + csin2(x) = 0 a=b=c= 0
1
2
∃M∈R:∀n∈N, un6M
∀x1∈R,∀x2∈R, x1< x2=⇒f(x1)6f(x2)
f I ∀x∈I, f0(x) = 0 f I
E={f∈D(R)| ∀x∈R, f0(x)=2xf(x)}F={x7→ kex2|k∈R}
D(R)R
E=F
F⊂E
f∈E g :x7→ e−x2f(x)R
R
Ea= ([−1,2] ∪[3,5]) \[1,3]
Eb=1
3;2 5∪3 5;3 4∩1 2;2 3
Ec=1
3;2 5∪3 5;3 4∩1 2;2 3

M(x, y) (x, y)∈D
D= [−1,2] ×[1,3]
D= ([−2,−1] ∪[0,2]) ×[1,3]
D= ([−1,2] ×[1,+∞[) ∪[−1,0] ×[−1,3]
D= ([−1,2] ×[1,+∞[) ∩([−∞; 1[×[2,3])
(xn) (yn)x0= 1 y0= 1 n
xn+1 =xn+yn
yn+1 = 2xn+yn
n xn>1yn>1
n P (n) : ” 2n>(n+ 1)2”
n>2P(n) =⇒P(n+ 1)
n P (n)
u0= 0 u1= 1 n un+2 = 4un+1 −3un
n∈Nun=3n−1
2
(un)n>1u1= 1 n un+1 =1
n+ 1 +n+ 2 n+ 1 un
n unn