Esercizi svolti di analisi reale e complessa - Dipartimento di ...
Esercizi svolti di analisi reale e complessa - Dipartimento di ...
Esercizi svolti di analisi reale e complessa - Dipartimento di ...
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sup x∈(a,b) |um(x) − un(x)| < ε <br />
sup x∈(a,b) |um(x) − un(x)| = sup x∈[a,b] |um(x) − un(x)|<br />
{xk}k≥2 = {a + b−a<br />
k }k≥2 ⊆ (a, b) <br />
limk→∞ xk = a un <br />
limk→∞ |un(xk) − um(xk)| = |un(a) − um(a)| <br />
|un(a) − um(a)| ≤ sup x∈(a,b) |um(x) − un(x)| <br />
b <br />
(a, b) <br />
[a, b] <br />
{un(a)}<br />
<br />
⎧<br />
⎨<br />
fn(x) =<br />
⎩<br />
<br />
2nx x ∈ [0, 1<br />
−2nx + 2 x ∈ [ 1<br />
2n ]<br />
1<br />
2n , n ]<br />
0 x ∈ [0, 1<br />
n ]<br />
x ≤ 0 fn(x) ≡ 0 ∀n x > 0 n0 = [ 1<br />
x ] + 1<br />
∀ n ≥ n0 fn(x) = 0<br />
∀ n ≥ 1 sup [0,1] |fn(x)| = 1<br />
lim 1<br />
0 fn = lim 1<br />
2n = 0 = 1<br />
0 f<br />
<br />
⎧<br />
⎨<br />
fn(x) =<br />
⎩<br />
4n2x x ∈ [0, 1<br />
2n ]<br />
−4n2x + 4n x ∈ [ 1 1<br />
2n , n ]<br />
0 x ∈ [ 1<br />
n , 1]<br />
<br />
<br />
1 [x(<br />
gn(x) =<br />
n − x)]k+1 x ∈ [0, 1<br />
n ]<br />
0 fn = gn<br />
1<br />
0 gn<br />
(−e α , e α )<br />
K ⊂ (−e α , e α )<br />
α ≤ 0<br />
• (1, +∞)<br />
• [a, +∞) a > 1