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342
CHAPTER 3. REALVALUED FUNCTIONS: DIFFERENTIATION
(b) If it is true (given
α
and
β
) for a given pair (
x,y
), then it is also
true for the pair (
t
1
/α
x,t
1
/β
y
) (verify this!), and so we can
assume without loss of generality that
xy
= 1
(c) Prove the inequality in this case by minimizing
f
(
x,y
) =
1
α
x
α
+
1
β
y
β
over the hyperbola
xy
= 1
.
13. Here is a somewhat diFerent proof of Theorem
3.6.6
, based on an
idea of Daniel Reem [
44
]. Suppose
S
⊂
R
3
is compact.
(a) Show that for every integer
k
= 1
,
2
,...
there is a
fnite
subset
S
k
⊂
S
such that for every point
x
∈
S
there is at least one
point in
S
k
whose coordinates diFer from those of
x
by at most
10
−
k
. In particular, for every
x
∈
S
there is a sequence of points
{
x
k
}
∞
k
=1
such that
x
k
∈
S
k
for
k
= 1
,...
and
x
= lim
x
k
.
(b) Show that these sets can be picked to be nested:
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
 Fall '08
 ALL
 Calculus

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