PROOF OF VARIOUS LIMIT PROPERTIES
In this section we are going to prove several of the fundamental facts and properties about limits which we saw previously. Before proceeding along with any of the proofs we must note that several of the proofs utilize the precise definition of the limit and this is assumed that not you have only read that section but that you have a quite good feel for doing that type of proof. If you are not comfortable with using the definition of the limit to prove limits you will get several of the proofs in this section, are not easy to follow.
The proofs which we'll be doing here will not be fairly as detailed as in the precise definition of the limit section. The "proofs" that we did in that section first did some work to find a guess for the d and then we verified the guess. The reality is that frequently the work to find the guess is not demonstrated and the guess for d is just written down and after that verified. For the proofs under this section where a d is really selected we'll do this that way. To make issues worse, in some of the proofs in this section work extremely different from those which were in the limit definition section.
Therefore, with that out of the way, let's find to the proofs.