Continuity at infinity

Usually \(f(x)\to L\text{ as }x\to \infty\) is defined in an epsilon N fashion. With a projective point of view, infinity becomes a point among the usual numbers, and the definitions of convergence at finite and infinite points can be unified. What strikes me is that this means that if f(x) converges as x goes to infinity, then f can be continuously extended at infinity.

Cauchy–Schwarz

Finally I understand the Cauchy–Schwarz inequality! No induction needed.

Theorem
\(a, b\in \mathrm{Rat}^n\)
\(n\in\mathrm{Nat}\)
                     
\((a\bullet b)^2 \leq \mathop{\rm Q}(a)\mathop{\rm Q}(b)\)

Proof
Recall that by definition
\[a\bullet b=\sum_{i=1}^n a_ib_i\]\[\mathop{\rm Q}(a)=\sum_{i=1}^n a_i^2\]\[\mathop{\rm Q}(b)=\sum_{i=1}^n b_i^2.\]
The claim, reformulated as
\[\left(\sum_{i=1}^n a_ib_i\right)^2 \leq \left(\sum_{i=1}^n a_i^2\right)\left(\sum_{i=1}^n b_i^2\right),\]
is equivalent to (by distributing terms)
\[\left(\sum_{i=1}^n a_i^2b_i^2\right)+\sum_{1\leq i<j\leq n}2a_ib_ia_jb_j \leq \left(\sum_{i=1}^n a_i^2b_i^2\right)+\sum_{1\leq i<j\leq n}a_i^2b_j^2+a_j^2b_i^2\]
and (by collecting terms)
\[0 \leq \sum_{1\leq i<j\leq n}a_i^2b_j^2+a_j^2b_i^2-2a_ib_ia_jb_j = \sum_{1\leq i<j\leq n}(a_ib_j-a_jb_i)^2.\]
The last statement is true because squares are positive and sums of positive numbers are positive.

A spiral

Let \(\mathbb{O}\) be the countable set of all infinite rational number sequences which are ultimately but not identically zero, and let \(H\) contain those elements in \(\mathbb{O}\) whose last nonzero element is positive. With respect to \(\mathbb{O}\), define \(H^\complement\). Then \(f(x) = -x\) is a bijection from \(H\) to \(H^\complement\).

Also, \(H\) looks like a spiral. To see that, require the last nonzero element to be the first, then the second, and then the third one, and watch this sequence of subsets twist into a new dimension at each step.

That's really something! A spiral whose reflection in the origin is its complement.