This is for an Insights article: Bivariate induction proof using Calc3

[benorin]

Homework Statement

https://www.physicsforums.com/insights/a-novel-technique-of-calculating-unit-hypercube-integrals/#Images-of-the-2-d-versions-of-the-some-of-the-sequence-of-sets-S-N2-and-the-geometry-of-the-next-headings-method-of-finding-extrema

Relevant Equations

The method of bivariate induction is laid out in the article, I just want someone to check my work, I used Lagrange Multipliers to find extrema of the sequence of sets to prove they are nested but not sure if that's enough, or do I need convexity too?

Link to my insight Article it's right where I need you to start checking, read the above boxes to, check out the picture to see examples of the kind of sequence of sets we are dealing with. I need you to read the section jusr below the first picture entitled "3.0.2 Lemma 2.1: Nesting Property of $S_N^n$".

 

[andrewkirk]

Do you need proper containment ($\subset$) in your proposition or would you be happy with just containment ($\subseteq$)?

If the latter, I think you can make the proof simpler by using P(n=1, N=1) as your base case rather than P(n=2, N=1). Then our base proposition is in a one-dimensional space:
$$P(1,1) \equiv S_2^1\subseteq S_1^1
\equiv \left(\left(\frac xb\right)^4 \leq 1 \Rightarrow \left(\frac xb\right)^2 \leq 1 \right)$$
which is easy to prove. In fact we can easily show that
$$\forall N\in \mathbb Z^+\ \ S_N^1=[-b,b]$$
from which it follows that P(1,N) is true for all positive integers N.

So then we just need to do induction over $n$.

 

[Prof B]

For a long time, mathematicians wrote $A \subset B$ to mean "A is a subset of B". It's easier to write than the other one. "Proper subset" arises so rarely that it can be written out in words when needed. As such, it has never needed its own symbol.

With the advent of computer typesetting the more complicated symbol $\subseteq$ became popular. I think it has overtaken $\subset$ in popularity. Let's say that the probability that a writer uses $\subset$ to represent the subset relation is p. Let's say that the probability that a math formula involving some kind of subsets actually involves proper subsets is q. (Choose p and q according to your personal beliefs, or do your own research, as they say. Personally, I choose p=0.1 and q = 0.001.) A simple application of Bayes rule will tell you what $\subset$ usually means.

 

[benorin]

In case you’re curious this is what the n=2 (ie 2-dimensional family of sets looks like