0.1 To the student:

0.1 To the student:

Since I give you these lecture notes, you might ask why should I go to class?
or why should I pay attention in class?. Well, analysis is not something that
comes easy to most people. I would highly recommend that you read ahead
a little bit and come to class already having read the material. Then you can
use the lecture time to fill in any gaps in your understanding and to make sure
that you get the “big picture”. Since you don’t have to spend your class time
copying everything from the board, you can actually spend more of the class
time thinking about the material!
Warning: There will be things discussed in class that are NOT in these
notes, so you should copy those down. I would suggest that you use the reverse
sides of the pages to write further notes, draw the diagrams, add my comments
about “why” something is true, add your own thoughts and insights about the
material, etc.

Following along with the proofs in the class, trying to see the “big picture”
will help you (the student) to start to see patterns in why one approach is used
in one proof but another approach is used in another.
Proving theorems is important because it provides the element of rigor necessary
in a mathematics class. However, this is not the only (or most important)
reason for proofs. We prove things not only to make sure that they are indeed
correct, but also to make sure that we understand why they work as they do.
Most often, the best way to approach a proof of some statement is first to develop
an intuitive, rough idea as to why the statement is true. This is often done
by drawing some pictures or diagrams or using analogies with similar (previous)
situations. However, this is just the first step. We must next turn our vague
intuition into a concrete proof, using the language of mathematics, including all
the definitions and theorems at our disposal.
Constructing proofs is quite different from solving problems in calculus.



Many calculus problems are purely computational – once you know the techniques
or pattern, you replicate this pattern (with minor variations) for all
similar problems. However, proof courses don’t usually offer these types of
computational questions. Often each question in these notes will seem to be
an individual question totally different from all the other questions. There is
usually no complete set of question types from which to draw examples to follow.
So how do you approach a given problem? First, make sure that you understand
all the terms in the question. Read the definitions. Look at any examples
in the notes (or from your class notes) and see what examples have been given
(for example, if the question is about sequences, look at various examples of sequences).
What you should look for is to see if you understand how the formal
definition fits with the examples and what are the range of possible behaviors.
Next see if you can come up with some example that violates the statement
that you are trying to prove (for example, if you are trying to prove that all
bounded sequences have an increasing or decreasing subsequence, try to find
a bounded sequence which has neither an increasing nor a decreasing subsequence).
If you can see why you cannot come up with such a counter-example,
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