, can beexpressed as 
, for some, 
, with 
, and
some, 
, with
.
Section 5.3
Exercise #32
We want to show that the sum,
, can beexpressed as
, for some,
, with , and
some, , with
.
The approach is similar to what is used inthe book
on page 200 to justify what follows line (6), using theintermediate value
theorem. However, in this unequal spacingcase, the application of the intermediate
value theorem is moredifficult to make.
Each of the finitely many (n) terms in sum
, arefunctions of two variables, 
, where
. The
function is
theproduct of two continuous functions. Specifically, 
, where
and
are eachcontinuous
functions. As the finite sum and product of continuousfunctions is a continuous
function, we may apply the intermediatevalue theorem to entire sum,
.
First, we apply the intermediate valuetheorem to the
sum on the factor,
. The theorem
implies thatfor some,
, 
,
In this case, the intermediate valuetheorem applies
only with the stipulation that
, andnoticing that the width 
satisfies 
.
Thetheorem
may be applied only because 
lies within the range ofvalues of the sum
.
Next, apply the intermediate value theoremagain --
this time on the factor 
in 
.
Now, we have thedesired result,
as the theorem gives that there is a value
-- with chosen
so that 
for
which
.
Again, thetheorem may
only be applied because the desired result
iswithin the
range of values of
.