This week we went over the epsilon-delta proof once again.
While we have covered the material before, this time, it was more formal and
the example was more complex. I also had an interesting conversation with the
Professor after class talking about the difference approaching something from a
rigorous perspective and approaching it from an intuitive perspective. Both
elements are required to write the proof but both require practice. In my own
experience, I know that I have trouble translating my conceptual understanding
of what are epsilon and delta into mathematical notation.
Currently, this is my understanding of the overall approach
to these kinds of proofs:
You are up against a
challenge of an opponent who is trying to “screw you over”. However, you have
the advantage of being able to make your move after he/she has made theirs. In
other words, if someone gives you an epsilon range you can always pick a delta
that fits that range. You can do it for any range that you are given and the epsilon
and delta have always been restricted to positive real numbers. This has been
the foundation for any epsilon/delta proof that we have encountered.
The difficulties come with solving the specific problem that
comes with solving epsilon/delta proofs.
For example, having the requirements
that |x – 3| < d Þ |x2
– 32| < e, requires techniques such as factoring, telescoping and
the use of the triangle inequality. Furthermore, it requires choosing a delta
that makes your overall problem simpler when you have to do addition at the
end. While in hindsight this problem seems straight-forward, I had difficulties
at the beginning of how to move forward.
I am going to spend the next week, trying to develop further
intuition of how to approach the problems, as well as review some basic
mathematical rules and techniques that my brain has forgotten. Hopefully with
these tools, the next time I encounter an epsilon/delta proof, I will be able
to approach it without feeling overwhelmed.
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