Semaine 16

A, B RA⊂Binf A, sup A, inf B, sup B
(un)n∈Nn∈Nvn= sup{upp≥n}vn= inf{upp≥n}
x∈Rn∈N∗
n−1
k=0
E x +k
n=E(nx).
k1 (αk)k=1,...,n 0< α1<··· < αnx∈]0; 1[ k∈ {1, . . . , n}
fk(x) = xαk(fk)k∈{1,...,n}
A, B RA∪Bsup(A∪B) = max(sup(A),sup(B))
inf (x1+x2+··· +xn)1
x1
+1
x2
+··· +1
xn
x1, . . . , xn>0,
n
a≤b card([a, b]∩Z) = E(b) + E(1 −a)a∈Za /∈Z
E, F f ∈ L(E, F )
Ker(f)⊂Ker(f2)
Ker(f) = Ker(f2)Im(f)∩Ker(f) = {0}
A, B RA+B={a+b(a, b)∈A×B}A+B
sup(A+B) = sup(A) + sup(B)
inf (−1)n+1
n+ 1 n∈N.
n∈N∗
(an, bn)∈N∗(2 + √3)n=an+bn√3 3b2
n=a2 n−1
E((2 + √3)n)
Rn[X]n U ∈ L(Rn[X]) U(P)(X) = P(X+ 1) P∈Rn[X]
U(1, X, X2, . . . , Xn)
A, B RA⊂Binf A, sup A, inf B, sup B
(un)n∈Nn∈Nvn= sup{upp≥n}vn= inf{upp≥n}
x∈Rn∈N∗
n−1
k=0
E x +k
n=E(nx).
k1 (αk)k=1,...,n 0< α1<··· < αnx∈]0; 1[ k∈ {1, . . . , n}
fk(x) = xαk(fk)k∈{1,...,n}
A, B RA∪Bsup(A∪B) = max(sup(A),sup(B))
inf (x1+x2+··· +xn)1
x1
+1
x2
+··· +1
xn
x1, . . . , xn>0,
n
a≤b card([a, b]∩Z) = E(b) + E(1 −a)a∈Za /∈Z
E, F f ∈ L(E, F )
Ker(f)⊂Ker(f2)
Ker(f) = Ker(f2)Im(f)∩Ker(f) = {0}
A, B RA+B={a+b(a, b)∈A×B}A+B
sup(A+B) = sup(A) + sup(B)
inf (−1)n+1
n+ 1 n∈N.
n∈N∗
(an, bn)∈N∗(2 + √3)n=an+bn√3 3b2
n=a2 n−1
E((2 + √3)n)
Rn[X]n U ∈ L(Rn[X]) U(P)(X) = P(X+ 1) P∈Rn[X]
U(1, X, X2, . . . , Xn)