The following three theorems give conditions under which limits may be interchanged. The first interchanges the operations limn®¥ and limx® x0, the last two interchange integration or differentiation with the limit of a sequence of functions. The first theorem is a generalization of our result on the completeness of C[a,b].
Theorem. Suppose that W is a metric space and x0 limit point. If {fn} is a uniformly Cauchy sequence of functions defined on W\x0 and each fn has limit Ln at x0, then
Homework (due Monday March 30) Prove this result. [Hint: Parts 2) and 3) are both e/3 proofs.]
Theorem. Suppose that {fn} is a uniformly Cauchy sequence of functions on [a,b] which are Riemann-Stieltjes integrable with respect to a nondecreasing function a, then
Proof. For each x Î [a,b] the sequence {fn(x)} is Cauchy and has a limit which we call f(x). An e/2 proof shows that in fact fn® f uniformly. To see that f is Riemann-Stieltjes integrable with respect a, we use the integrabilty condition (*), that is for each e > 0, we need to show that we can find partitions so that the upper and lower Riemann-Stieltjes sums are arbitrarily close. Let e > 0 and pick N such that
fn - e £ f £ fn + e
if n ³ N. But by properties of upper and lower sums, we see that any partition P,
where we denote D a: = a(b)-a(a). By using the Riemann-Stieltjes integrability of fN, we may choose a partition P such that the left side of inequality (1) is within (1+2 Da)e of the right hand side of inequality (3) for n = N. Since e > 0 was arbitrary, f is Riemann-Stieltjes integrable with respect to a. These same inequalities also show that the integral òab f da satisfies
L(P;fn,a) - e Da £ òab f da £ U(P;fn,a) + e Da
for all partitions P. Hence
|òab fn da- òab f da| £ 2e Da
for n ³ N. [¯]
Theorem. Suppose that {fn} is a sequence of differentiable functions defined on [a,b] which converges for some x0 Î [a,b] and whose derivatives are continuous and uniformly Cauchy, then
Proof. We apply the previous theorem to the special case of Riemann integrable functions. Again, the fn¢ converge uniformly to a continuous function g. Let y0: = limn®¥ fn(x0), then define
f(x) : = òx0x g dx + y0,
then the fundamental theorem of calculus implies that
fn(x) : = òx0x fn¢ dx + fn(x0).
The desired result is established by taking the limit as n®¥ in
|fn(x)-f(x)| £ |òx0x (fn¢ -g) dx| + |fn(x0)-y0|
and applying the previous theorem. [¯]