Lecture 16 : Integral from upper and lower sums [Section 16.1]
16.1.7
Theorem:
Let be a sequence of refinement
partitions of . Then the following hold:
(i)
The sequence is monotonically decreasing and is bounded below by .
(ii)
The sequence is monotonically increasing and is bounded above by .
(iii)
Both and are convergent sequences. In fact, for is continuous,
and the limit is independent of the sequence of refinement partition of .
Proof:
Proofs of (i) and (ii) follow directly from the lemma 16.1.6. Thus, by the completeness property of , both exist. That for a continuous function, both these sequences converge to a common limit, and that limit is independent of the sequence of refinement partitions of is technical, and we assume this.
be a sequence of refinement 
. Then the following hold: 
is monotonically decreasing and is bounded below by 
. 
is monotonically increasing and is bounded above by 
. 
and 
are convergent sequences. In fact, for 
is continuous, 
of refinement partition of 
. 
, both 
exist. That for a continuous function, both these sequences converge to a common limit, and that limit is independent of the sequence 
of refinement partitions of 
is technical, and we assume this. 
