un programme ambitieux

♠
π
(un)nk| · |
Q R C
∀ε > 0,∃N∈N∀m, n >N, |un−um|6ε.
ε
(un)nk
(un)n⇒(un)n⇒(un)n.
k=R C ⇔
k=Q
ezz∈C
Pn
k=0
xk
k!n
xR+
z∈CPn
k=0
zk
k!n
C
exp(z)
♠z, w ∈C,exp(z+w) = exp(z) exp(w)
z∈C,exp(z) = exp(z)
n∈Nexp(n) = ene:= exp(1)
ez:= exp(z)
cos,sin π
θ∈R|eiθ |= 1
cos θ:= R(eiθ ) sin θ:= I(eiθ )
a, b ∈R,cos(a+b) = cos acos b−sin asin b
sin(a+b) = sin acos b+ sin bcos a

∃limx→0sin x
x= 1 ∃limx→0cos x= 1 cos
sin R
♠cos R+sin
>0 +∞cos
♠ {x∈R+cos x= 0}π
R→U, θ 7→ eiθ
R
C(Q) (rn)n∈QN
ϕ:C(Q)→R,(rn)n7→ limnrn
ϕ
ϕ((rn)n) = ϕ((sn)n) limn(rn−sn)=0
R(rn)n∈
QN0
∀(rn)n,(sn)n∈ C(Q),(rn)n∼(sn)n⇔ ∃ lim(rn−sn)=0.
C(Q)
R
♠
♠
♠Q→R
♠R6(R,6)
x, y ∈R∗
+,∃n∈Nnx >y
R
R Q
d(x, y) := max(x−y, y −x)
Q R
sin x= limn→+∞Pn
k=0
(−x)k
(2k+1)!
Q
p
Qp
C