steps that show the epsilon and delta proof (multivariable)

Question

Labelled below from A to K are the steps that show (5 steps) (sorry it is a picture and not written in latex)

I picked the following:D,C,F,G,B I was wondering if that is correct or if there is another way.

Answer 1

The only sequence of steps that's both mathematically and grammatically correct is D,G,F,B,E in that order.

D.

For every $\epsilon>0$,

G.

we have $\left|\frac{y^2}{\sqrt{x^2+y^2}}-0\right| = \frac{y^2}{\sqrt{x^2+y^2}} \le \frac{x^2+y^2}{\sqrt{x^2+y^2}} = \sqrt{x^2+y^2}$.

F.

It follows that $\left|\frac{y^2}{\sqrt{x^2+y^2}}-0\right| < \delta$ whenever $\sqrt{x^2+y^2} < \delta$ for some number $\delta>0$.

B.

Choose $\delta$ such that $\delta = \epsilon$.

E.

For this choice, $\left|\frac{y^2}{\sqrt{x^2+y^2}} - 0\right| < \epsilon$.

This is very nearly the same choice of steps you have, except that:

• you're missing step E, which is the only step with the necessary conclusion: that $|f(x,y) - L| < \epsilon.$
• you have the extraneous step C, which doesn't fit in with the usual $(\epsilon,\delta)$ proof structure (we are given $\epsilon$, but we get to pick $\delta$ based on $\epsilon$) and also is redundant given that you have step B, where we pick a $\delta$.

I wouldn't call either of these serious omissions; this problem requires some amount of reading the author's mind to figure out what the steps are intended to mean out of context.

There's a lot of freedom in how to order these steps. Mathematically, I want to put G before F: the logic here is that we prove $A \le B$ (this is in step G), and therefore $A < \delta$ whenever $B < \delta$ (this is in step F). This is also the only aspect of the order where we disagree.

My other reasons for ordering these steps are non-mathematical (and wouldn't apply if we got to pick the wording of the proof, the way we would if we were really writing it):

• G should go immediately after D, because they are sentence fragments that form a sentence together.
• B should go after F, because the way F is worded, we haven't chosen a $\delta$ yet.
• E should go after B, because it begins "For this choice", so it should happen after we make a choice, and B says "Choose".

In an actual proof, it wouldn't be unreasonable to put the equivalent of step B as early as immediately after D, for example. But with this specific wording of the steps, doing so wouldn't make sense.