This week we took another detour before getting to the construction of the real numbers to discuss limits.Week_8___Limits
Just a quick update to correct an error in a proof from last week.Week_7___Continuous_Functions_on_the_Rationals (1)
This week we took a slight detour before getting to the real numbers to discuss proving the continuity of rational-valued functions using delta-epsilon arguments.
Since this week’s classes were spent discussing proof, how to come up with ideas for proving something, and how to get those ideas down onto paper, I decided to forge ahead and start next week’s material by reading ahead in the textbook and taking notes on some key ideas regarding the set of real numbers. This week’s post includes these key ideas, along with a few proofs that I worked on alongside the reading that are most relevant to real analysis.Week_6___The_Real_Numbers
This week, I worked on the proof from the previous week. Upon completion of this proof, I used it to work on proofs for the other exercises for irrational numbers. For these exercises, based on what we discussed in class, I primarily focused on explaining the concepts necessary to establish the proof, rather than just creating a long list of algebraic manipulations. This week should complete my investigation of the irrational numbers.Week_5___sqrt_2__and_the_Irrational_Numbers (4)
This week I struggled with a proof regarding the convergence of the continued fraction for approximating sqrt(2), finding that statement that the exercise asserted was actually false. Since this proof was required to complete further exercises, this week’s work is shorter than the previous; I have only included two short proofs this week regarding continued fractions.Week_4___sqrt_2__and_the_Irrational_Numbers (4)
This week I investigated the set of irrational numbers, translating some common techniques used to prove the irrationality of sqrt(2) to prove the irrationality of other numbers. I also explored a proof of a method of approximating sqrt(2), which brings my investigation into the realm of mathematics known as “analysis”Week_3___sqrt_2__and_the_Irrational_Numbers
Here is my work for week 2, which I have added to my work for week 1. This week, I worked on filling in some holes, specifically with regards to constructing the set of integers, proving the Archimedean ordering property, and adding a proof regarding limits of sequences.Week_2___The_Field_of_Rational_Numbers (3)
Here are my findings for week 1, in LaTeX form. Many relatively simple proofs were completed, and I found proof by induction to be very necessary for the initial proofs regarding the set of natural numbers.Week_1___The_Field_of_Rational_Numbers (1)
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