The Art and Craft of Problem Solving

318 CHAPTER 9 CALCULUS

Convergence

We say that the real-valued sequence (an) converges to the limit L if

lim an =L.

n->oo

What does this mean? That if we pick an arbitrary distance e > 0, then eventually, and

forever after, the ai will get within e of L. More specifically, for any e >^0 (think of e

as a really tiny number), there is an integer N (think of it as a really huge number, one

that depends on e) such that all of the numbers

lie within e of L. In other words, for all n 2 N,

If the context is clear, we may use the abbreviation an ---t L for lim an = L.

n->oo

In practice, there are several possible methods of showing that a given sequence

converges to a limit.

1. Draw pictures whenever possible. Pictures rarely supply rigor, but often fur­
   nish the key ideas that make an argument both lucid and correct.

2. Somehow guess the limit L, and then show that the ai get arbitrarily close to L.

3. Show that the ai eventually get arbitrarily close to one another. More precisely,

a sequence (an) possesses the Cauchy property if for any (very tiny) e > 0

there is a (huge) N such that

lam-ani < e

for all m,n 2 N. If a sequence of real numbers has the Cauchy property, it

converges.^2 The Cauchy property is often fairly easy to verify, but the disad­

vantage is that one doesn't get any information about the actual limiting value

of the sequence.

4. Show that the sequence is bounded and monotonic. A sequence (an) is

bounded if there is a finite number B such that I an I ::; B for all n. The sequence

is monotonic if it is either non-increasing or non-decreasing. For example, (an)

is monotonically non-increasing if an+l ::; an for all n.

Bounded monotonic sequences are good, because they always converge. To

see this, argue by contradiction: if the sequence did not converge, it would not

have the Cauchy property, etc. But please note: the limit of the sequence need

not be the bound B! Construct an example to make sure you understand this.

5. The Squeeze Principle. Show that the terms ofthe sequence are bounded above
   and below by the terms of two convergent sequences that converge to the same
   limit. For example, suppose that for all n, we have

0< Xn < (0.^9 t.

(^2) See [36] for more information about this and other "foundational" issues regarding the real numbers.