The Art and Craft of Problem Solving

9.2 CONVERGENCE AND CONTINUITY 325

Uniform continuity is just what we need to complete our proof of the FTC.

Example 9.2.7 Show that

lim

(E( t))

.11-+0 .1t =^0 ,

where E ( t) was the shaded area in the diagram on page 317.

Solution: Since f is defined on the closed interval [a,b] and is continuous, it is

uniformly continuous. Pick e > 0. By uniform continuity, there is a small enough .1t

so that no matter what t is, the distance between f( t) and f ( t + .1t) is less than e. In

other words,

If(t+.1t) -f(t) 1 < e.

Thus the area of the "error" E ( t) (in absolute value) is at most e. .1t, and hence

I E(t) 1 e ·.1t

--<--=e.

.1t .1t

In other words, no matter how small we pick e, we can pick a small enough .1t to

guarantee that E ( t) /.1t is less than e. Hence the limit is 0, and we have proven the

FTC. •

Uniform continuity, as you see, is a powerful technical tool. But remember, the

crux idea in our proof of the FTC was to picture a moving curtain. This simple picture

is easy to remember and immediately leads to a one-sentence "proof' that lacks tech­

nical details. The details are important, but the picture--or at least, the idea behind the

picture-is fundamental.

Problems and Exercises

The problems in this chapter are among the most challenging in the book, because "calculus" is a
huge, open-ended subject that quickly speeds off into higher mathematics. We highly recommend
that you supplement your reading by perusing some of the calculus texts mentioned on 357.

9.2.8 Arnie plays Scrabble with Betty. He has kept

track of the percentage of games that he has won since

they began playing. He told his friend Carla, "Some

time ago, my win percentage was less than p percent,

but as of today, it is more than p percent." Carla said, "I

can prove that at some time in the past, your win per­

centage was exactly p percent." Assuming that Carla's

assertion is correct, what can you assert about p? (In­

spired by a 2004 Putnam problem.)

9.2.9 Define the sequence (an) by al = 1 and an =

1 + 1/ an-I for n 2': 1. Discuss the convergence of this

sequence.

both converge. What can you say about the conver­

gence of the sequence (bn)?

9.2. 11 Interpret the meaning of

9.2. 12 Fix a > I, and consider the sequence (xn)n�O

defined by Xo > va and

Xn+ a

Xn+1 =

Xn +^1 '

n = 0, 1,2, ....

9.2.10 Suppose that the sequences (an) and (an/bn) Does this sequence converge, and if so, to what? Re­

late this to Example 9.2.2 on page 320.