The following three theorems give conditions under which limits may be interchanged. The first interchanges the operations lim_n®¥ and lim_{x® 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 {f_n} is a uniformly Cauchy sequence of functions defined on W\x₀ and each f_n has limit L_n at x₀, then

1. there exists a function f such that f_n ® f uniformly on W\x₀.
2. {L_n} is Cauchy. Call its limit L.
3. f has a limit at x₀ and it is equal to L.

Homework (due Monday March 30) Prove this result. [Hint: Parts 2) and 3) are both e/3 proofs.]

Theorem.  Suppose that {f_n} is a uniformly Cauchy sequence of functions on [a,b] which are Riemann-Stieltjes integrable with respect to a nondecreasing function a, then

1. there exists a function f such that f_n ® f uniformly on [a,b],
2. f is Riemann-Stieltjes integrable with respect a,
3. lim_n®¥(ò_a^b f_n  da) exists and equals ò_a^b f  da.

Proof.  For each x Î [a,b] the sequence {f_n(x)} is Cauchy and has a limit which we call f(x). An e/2 proof shows that in fact f_n® 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

f_n - e £ f £ f_n + e

if n ³ N. But by properties of upper and lower sums, we see that any partition P,

1. L(P;f_n,a) - e Da £ L(P;f,a)
2. L(P;f,a) £ U(P;f,a)
3. U(P;f,a) £ U(P;f_n,a) + e Da

where we denote D a: = a(b)-a(a). By using the Riemann-Stieltjes integrability of f_N, 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 ò_a^b f  da satisfies

L(P;f_n,a) - e Da £ ò_a^b f  da £ U(P;f_n,a) + e Da

for all partitions P. Hence

|ò_a^b f_n  da- ò_a^b f  da| £ 2e Da

for n ³ N.   ^[¯]

Theorem.  Suppose that {f_n} 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

1. lim_{n® ¥} f_n¢ = g uniformly, for some continuous g,
2. the sequence {f_n} is uniformly Cauchy and so has some continuous function f as its limit.
3. f is differentiable with derivative g.

Proof.  We apply the previous theorem to the special case of Riemann integrable functions. Again, the f_n^¢ converge uniformly to a continuous function g. Let y₀: = lim_n®¥ f_n(x₀), then define

f(x) : = ò_x0^x g  dx + y₀,

then the fundamental theorem of calculus implies that

f_n(x) : = ò_x0^x f_n^¢  dx + f_n(x₀).

The desired result is established by taking the limit as n®¥ in

|f_n(x)-f(x)| £ |ò_x0^x (f_n^¢ -g)  dx| + |f_n(x₀)-y₀|

and applying the previous theorem.   ^[¯]