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In this section we are going to prove some of the basic
properties and facts about limits that we saw in the Limits
chapter. Before proceeding with any of
the proofs we should note that many of the proofs use the precise definition of the limit and it is assumed
that not only have you read that section but that you have a fairly good feel
for doing that kind of proof. If you’re
not very comfortable using the definition of the limit to prove limits you’ll
find many of the proofs in this section difficult to follow.
The proofs that we’ll be doing here will not be quite as
detailed as those in the precise definition of the limit section. The “proofs” that we did in that section
first did some work to get a guess for the 
and then we verified the guess. The reality is that often the work to get the
guess is not shown and the guess for 
is just written down and then
verified. For the proofs in this section
where a 
is actually chosen we’ll do it that way. To make matters worse, in some of the proofs
in this section work very differently from those that were in the limit
definition section.
So, with that out of the way, let’s get to the proofs.

Limit Properties
In the Limit Properties
section we gave several properties of limits. We’ll prove most of them
here. First, let’s recall the properties
here so we have them in front of us.
We’ll also be making a small change to the notation to make the proofs go
a little easier. Here are the properties
for reference purposes.
Note that we added values (K, L, etc.) to each of the limits to make the
proofs much easier. In these proofs
we’ll be using the fact that we know 
and 
we’ll use the definition of the limit to make
a statement about 
and 
which will then be used to prove what we
actually want to prove. When you see
these statements do not worry too much about why we chose them as we did. The reason will become apparent once the
proof is done.
Also, we’re not going to be doing the proofs in the order
they are written above. Some of the
proofs will be easier if we’ve got some of the others proved first.
Proof of 7
Proof of 1
Proof of 2
Note that we’ll need something called the triangle inequality in this proof. The triangle inequality states that,
Here’s the actual proof.