Esercizi svolti di analisi reale e complessa - Dipartimento di ...

•
x y ℓ1
x + y1 =
|xn + yn| ≤
n∈N
≤
(|xn| + |yn|) =
n∈N
=
|xn| +
|yn| =
n∈N
n∈N
= x1 + y1 .
{x (k) }
ℓ 1 ε > 0 N0 = N0(ε) > 0
k, h > N0
x (k) − x (h) 1 ≡
n∈N
n ∈ N
|x (k)
n − x (h)
n | ≤ ε .
|x (k)
n − x (h)
n | ≤ ε
{x (k)
n }k R
k +∞
xn
x = {xn}n .
x {x (k) }k
· 1
M > 0 k, h > N0
M
n=0
|x (k)
n − x (h)
n | ≤ ε ;
h → +∞
M
n=0
|x (k)
n − xn| ≤ ε
M
ε ≥
∞
n=0
|x (k)
n − xn| = x (k) − x1