My lemma is that if $P$ is a partition such that $U(f,P)-L(f,P)<(b-a)/m$ for some $m$ natural, then there must exist ad index $i$ for which $M_i-m_i<1/m$
Let $\dim V=n$ and $f$ nilpotent. Then there exists $k$ such that $f^k=0$,. Consider the set $K=\{k:f^k=0\}$. By the well ordering principle $K$ has a least element $k'$. We wish to prove $k'\leq n$
If the Cantor set consists of all base 3 "decimals" with only 0 and 2, then isn't the collection of all such "decimals" which terminate a countable dense subset?
@PeterTamaroff Hmm. Without determinants, you will get into some lengthy argument of linear independence. I am not sure how. Let $L b_i = a_1i b_1 + a_2i b_2 + \cdots + a_ni b_n$ where $b$ are the basis of course. Something like that, which are basically hypervectors. Let me think.
The fourier coefficients for nonzero $n$ are
$$\begin{eqnarray}
\hat f(n)
&=& \frac{1}{2 \pi} \int_0^{2 \pi} (\pi-t) e^{-i n t} dt \\
&=& \frac{1}{2} \int_0^{2 \pi} e^{-i n t} dt - \frac{1}{2 \pi} \int_0^{2 \pi} t e^{-i n t} dt \\
&=& - \frac{1}{2 \pi} \left(\left[t \frac...
