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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1<br />
2<br />
− |xm| −<br />
<br />
1 <br />
<br />
2 − |ym| =<br />
=<br />
=<br />
=<br />
=<br />
=<br />
<br />
<br />
<br />
<br />
1<br />
2 − 1 + 1<br />
2m<br />
1<br />
1 + 1<br />
2m<br />
2m<br />
(1 + 2m) −<br />
−<br />
− 1<br />
1 + 1<br />
m<br />
1<br />
2 − 1 + 1<br />
=<br />
m<br />
(1 + m) =<br />
m<br />
(2m + 1)(m + 1) =<br />
2m2 1<br />
(2m + 1)(m + 1) 2m =<br />
m<br />
<br />
<br />
<br />
=<br />
2m 2<br />
(2m + 1)(m + 1) |xm − ym| .<br />
Lm <br />
xm ym <br />
Lm :=<br />
L ≥ 1<br />
<br />
<br />
<br />
<br />
<br />
sin<br />
n<br />
i=1<br />
xi<br />
<br />
<br />
<br />
<br />
−<br />
<br />
<br />
<br />
<br />
sin<br />
n<br />
i=1<br />
yi<br />
2m2 m→+∞<br />
−→ 1<br />
(2m + 1)(m + 1)<br />
<br />
<br />
<br />
<br />
≤<br />
<br />
<br />
<br />
<br />
sin<br />
<br />
n<br />
n <br />
<br />
xi − sin yi<br />
<br />
i=1<br />
i=1<br />
≤<br />
<br />
n n<br />
<br />
<br />
<br />
<br />
xi − yi<br />
<br />
<br />
i=1 i=1<br />
≤<br />
<br />
<br />
<br />
≤ <br />
x1<br />
n<br />
n<br />
n<br />
n<br />
<br />
<br />
<br />
xi − x1 yi + x1 yi − y1 yi<br />
<br />
i=2 i=2 i=2 i=2<br />
≤<br />
<br />
<br />
n n <br />
<br />
<br />
≤ |x1| xi − yi<br />
<br />
<br />
i=2 i=2<br />
+<br />
<br />
n <br />
<br />
yi<br />
<br />
i=2<br />
|x1 − y1| ≤<br />
<br />
n n<br />
<br />
<br />
<br />
<br />
≤ |x1 − y1| + xi − yi<br />
<br />
≤<br />
≤ . . . ≤<br />
<br />
| |x| − |y| | ≤ |x − y| x, y ∈ R n <br />
| sin t − sin s| ≤ |t − s| t, s ∈ R <br />
x1 ≤ √ n|x| x ∈ R n <br />
i=2<br />
i=2<br />
≤ |x1 − y1| + . . . + |xn − yn| =<br />
= x − y1 ≤<br />
≤ √ n |x − y| .