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

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 .

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.

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.

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.

Then

and we conclude that

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

i.e.

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

then

If

then

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

If then so

This shows that

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

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

(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