1549901369-Elements_of_Real_Analysis__Denlinger_

2.1 Basic Concepts: Convergence and Limits 61

SUMMARY: HOW TO PROVE lim xn = L

n-+oo

1. Let c > 0.


2. Find a real number r such that lxn - LI < c for all n 2: r.
   (This is what we did in Part (c) of Examples 2 .1. 5 and 2.1.7.)


3. Let no denote a ny natural number 2: r (found in Ste.p 2).
   (The Archimedean property guarantees the existence of this n 0 .)


4. Prove directly that for this value of no, n 2: no =? lxn - LI < c.
   (This is what we did in Examples 2.1.6 and 2.1.8.)

Note: Step 2 above, although of critical importance in finding n 0 , is not
considered part. of the proof of n--+ex> lim Xn = L. It is never included when the

proof is written up. It may be discarded once Step 4 is completed. In fact, step
4 is usually done by working Step 2 backwards, as demonstrated in Examples
2.1.6 and 2.1.8.

EXERCISE SET 2.1

l. Write out the first eight terms of each of the following sequences:

r (a) { ~2} (b) {(-l)n}

(c) {n~} (d) {(1 + *r}

( e) {sin ( mr)} (f) {cos ( m r)}

~g) { COS ( n 3 n) } (h) { n

2

~ 2n}

2. In each of the following exercises, a limit statement n-+oo li m Xn = L is given.
   In each case, answer the following questions:

(1) After how many t erms are we guaranteed that Xn is within. 01 of L?

(2) After how many terms are we guaranteed that Xn is an accurat e

approximation of L to within 3 decimal places?

(3) For arbitrary but unknown c > 0, after how many terms are we

guaranteed that Xn is within c of L?