I am working through this post from Terry Tao's blog.
Problem: Prove that if a binary (i.e. consisting of $0$ and $1$, or $\top$ and $\bot$) sequence $x_n$ is such that $\text{p-lim}(x_n) = 1$, then any sequence $y_n$ containing $x_n$ has $\text{p-lim}(y_n) = \text{p-lim}(x_n) = 1$.
My attempt at a proof. We know the following facts regarding $\text{p-lim}$:
- homomorphic: $\text{p-lim}(1) = 1$, $\text{p-lim}(Cx_n) = C\text{p-lim}(x_n)$ (for some $C \in \{0, 1\}$), $\text{p-lim}(x_n \oplus y_n) = \text{p-lim}(x_n) \oplus \text{p-lim}(y_n)$ (where $\oplus$ is the "or" operator), and $\text{p-lim}(x_n \otimes y_n) = \text{p-lim}(x_n) \otimes \text{p-lim}(y_n)$ (where $\otimes$ is the "and" operator)
- bounded: $\text{p-inf}(x_n) \leq \text{p-lim}(x_n) \leq \text{p-sup}(x_n)$
- non-principality: deletion of a finite number of elements from $x_n$ should not change its $\text{p-lim}$.
We do not know anything else about $\text{p-lim}$, so a combination of these facts must be used to prove the required.
Note that $x_n$ will never be finite, since it is an indicator sequence which runs over all the natural numbers, hence $y_n$ will never be finite either. So non-principality is unlikely to be a useful tool. The supremum and infimum of every sequence of binary numbers which is not $1, 1, 1, 1, 1, 1 \dots$ or $0, 0, 0, 0, 0, 0, \dots$ is trivially $0$ and $1$ respectively, so it is unlikely that boundedness can be used as a tool.
Let us try to use the fact that $\text{p-lim}$ is a homomorphism. Note that $y_n \oplus x_n = y_n$, since $x_n \subset y_n$ (in other words, $x_k = 1 \Rightarrow y_k = 1$, while $x_k = 0 \Rightarrow (y_k = 1 \vee y_k = 0)$). Then:
\begin{align*} &\quad\;\; x_n \oplus y_n = y_n \\ &\Rightarrow \text{p-lim}(x_n \oplus y_n) = \text{p-lim}(y_n) \\ &\Rightarrow \text{p-lim}(x_n) \oplus \text{p-lim}(y_n) = \text{p-lim}(y_n) \\ &\Rightarrow 1 \oplus \text{p-lim}(y_n) = \text{p-lim}(y_n) \\ &\{\text{1 is the zero of $\oplus$}\} \\ &\Rightarrow 1 = \text{p-lim}(y_n) \end{align*}
Is this proof correct? The following assumptions, I think, will need confirmation:
- On the space of binary numbers, is it okay to assume that $\oplus, \otimes$ are the analogues of $+, \times$, for the homomorphism property? I think it should be okay, because: 1) it makes things work out nicely (I think I can use similar logic to show that some of the other ultrafilter properties hold for the $\text{p-lim}$ over the binary numbers), and connects nicely to classical "predicate calculus" stuff; 2) $\oplus, \otimes$ can also be thought of as "addition/multiplication $\mathbb{Z}/2$")