Note that interior points of must belong to . Approachable points of need not belong to . Any interior point of is approachable from .

If is the closed interval then the points that are approachable from are exactly the points in , and the interior points of are the points in the open interval .

if

**1)**- is approachable from , and
**2)**- For every sequence in

Note that the value of (if it exists) has no influence on the meaning of . Also the `` " in (10.8) is a dummy variable, and can be replaced by any other symbol that has no assigned meaning.

does not exist. If , then the domain of consists of the single point , and is not approachable from . If we did not have condition

which would not be a good thing. (If there are no sequences in , then

is true, no matter what statement about is.)

In this course we will not care much about functions like .

for all .

Case 1: Suppose
. Let be a generic sequence in
such that . Then

Now, since , we have

so by the squeezing rule which is equivalent to

This proves (10.14) when .

Case 2: Suppose . The domain of the square root function is , and is approachable from this set.

Let be a sequence in such that . To show that , I'll use the definition of limit. Let . Then , so by the definition of convergence, there is an such that for all we have . Then for all we have and hence .

Many of our rules for limits of sequences have immediate corollaries as
rules
for limits of functions. For example, suppose are real valued functions with
and
. Suppose
and
. Let be a generic sequence in
such that . Then
is a
sequence in
and , so

Also is a sequence in and so

By the sum and product rules for sequences, for any

and

and thus we've proved that

and

Moreover if (so that ), and if for all (so that for all ), it follows from the quotient rule for sequences that

so that

Actually all of the results just claimed are not quite true as stated. For we have

and

but

The correct theorem is:

be real valued functions with and , and let . Suppose , and both exist. Then

Proof: Most of the theorem follows from the remarks made above. We will assume the remaining parts.

**i**- and both exist.
**ii**- is approachable from .
**iii**- There is a positive number such that
for all in .

Similiarly

Also for all , so it follows from the inequality rule for limits of sequences that , i.e. .

**i**- and both exist and are equal.
**ii**- is approachable from .
**iii**- There is a positive number such that
for all in

.