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# 6.4 Properties of Limits.

In this section I will state some basic properties of limits. All of the statements listed here as assumptions are, in fact, theorems that can be proved from the definition of limits. I am omitting the proofs because of lack of time, and because the results are so plausible that you will probably believe them without a proof.

6.35   Definition (Constant sequence.) If is a real number then the sequence all of whose terms are equal to is called a constant sequence

It is an immediate consequence of the definition of convergence that

for every real number . (If for all in then for all in so approximates with an error smaller than for all . .)

We have just proved

6.36   Theorem (Constant sequence rule.) If denotes a constant sequence of real numbers, then

6.37   Theorem (Null sequence rule.) Let be a positive rational number. Then

Proof: Let be a positive rational number, and let be a generic positive number. By the monotonicity of powers , we hwve

By the Archimedian property for there is an integer in such that

Then for all in

Thus .

6.38   Assumption (Sum rule for sequences.) Let and be convergent sequences of real numbers. Then

and

The sum rule is actually easy to prove, but I will not prove it. (You can probably supply a proof for it.)

Notice the hypothesis that and are convergent sequences. It is not true in general that

For example, the statement

is false, since

but neither of the limits or exist.

6.39   Assumption (Product rule for sequences.) Let and be convergent sequences. Then

An important special case of the product rule occurs when one of the sequences is constant: If is a real number, and is a convergent sequence, then

The intuitive content of the product rule is that if approximates very well, and approximates very well, then approximates very well. It is somewhat tricky to prove this for a reason that is illustrated by the following example.

According to Maple,

so approximates with 4 decimal accuracy. Let

and let

Then approximates with 4 decimal accuracy and approximates with 4 decimal accuracy. But

and

so does not approximate with an accuracy of even one decimal.

6.40   Assumption (Quotient rule for sequences.) Let and be convergent real sequences such that for all in and . Then

The hypotheses here are to be expected. If some term were zero, then would not be a sequence, and if were zero, then would not be defined.

6.41   Assumption (Inequality rule for sequences.) Let and be convergent sequences. Suppose there is an integer in such that

Then

The most common use of this rule is in situations where

and we conclude that

6.42   Assumption (Squeezing rule for sequences.) Let , , and be three real sequences. Suppose there is an integer in such that
 (6.43)

Suppose further, that and both converge to the same limit . Then also converges to .

If we knew that the middle sequence, in the squeezing rule was convergent, then we would be able to prove the squeezing rule from the inequality rule, since if all three sequences , and converge, then it follows from (6.43) that

i.e.

and hence . The power of the squeezing rule is that it allows us to conclude that a limit exists.

6.44   Definition (Translate of a sequence.) Let be a real sequence, and let . The sequence is called a translate of .

6.45   Example. If

then

If

then

6.46   Theorem (Translation rule for sequences.) Let be a convergent sequence of real numbers, and let be a positive integer. Then is convergent and

Proof: Suppose , and let be a generic element in . Then we can find an integer in such that

If then so

This shows that

6.47   Example. The sequence

is a translate of the sequence . Since it follows from the translation theorem that also.

6.48   Theorem (th root rule for sequences.) Let be a positive number then

Proof: Case 1: Suppose . Then

Case 2: Suppose , so that for all . Let be a generic positive number, and let be a generic element of . Since is strictly increasing on we have

 (6.49)

(In the last step I used the fact that if .) By the Archimedean property for there is an integer in such that

For all we have

Hence .

Case 3: Suppose . Then so by Case 2, we have

Thus, in all cases, we have

Next: 6.5 Illustrations of the Up: 6. Limits of Sequences Previous: 6.3 Convergence of Sequences   Index
Ray Mayer 2007-09-07