Basic Logic

I have written up a short essay that gives a brief and informal overview of the logical syntax used in mathematics. It covers the basic logical connectives and quantifiers, and gives a few examples. In particular, the final example (negation of the ε-δ definition of continuity) should be helpful in solving problem 2 in this week’s homework for Math 32AH. Here is a link to the essay: Basic Logic. As usual, let me know in the comments below if anything in the essay is incorrect or unclear.…

Math 131A Review Questions

I have compiled a few final review questions for real analysis this quarter. They are available at Questions Solutions Solutions are now posted as well. As usual, let me know in the comments below if you notice any errors or if anything is unclear.…

A smooth but not analytic function

In the current assignment for real analysis, we consider the following function $f(x) = \begin{cases} e^{-1/x^2} & x \neq 0\ 0 & x = 0. \end{cases}$ We are asked to show that $$f^{(n)}(0) = 0$$ for all $$n$$, and that the Taylor series for $$f$$ at $$0$$ converges to $$f(x)$$ only for $$x \neq 0$$. To compute the derivatives $$f^{(n)}(0)$$, it is easiest to use induction on $$n$$. For $$n = 1$$, \[ f'(0) = \lim_{x \to 0} \frac{f(x) – f(0)…

I didn’t finish a careful proof of the triangle inequality in class, so I am presenting a more polished argument here. I abandoned the text’s argument in favor of what I hope is a more intuitive (albeit longer) proof. I’ve also had a number of requests for hints on a couple homework problems, specifically numbers 3.7 and 3.8. So here are my comments on those problems: Problem 3.7 This problem asks us to prove that $$|b| 0$$, for otherwise the assumption \(|b| Problem 3.…