I just found that the trace is the \( n-1 \)th coefficient of the characteristic polynomial.
As we want to show that the solutions of the sequence of characteristic polynomials converge to \( \{u^k: 0\le k\} \), maybe it suffices already to show that the coefficients of the sequence of characteristic polynomials converge to the coefficients of \( \Pi_{k=0}^n (x-u^k) \).
And indeed the \( n-1 \)th coefficient of \( \Pi_{k=0}^n (x-u^k) \) is \( \sum_{k=0}^n -u^k \), which is \( \frac{1}{1-u} \) in the limit.
So we have already one element of a proof, i.e.
the difference of the \( n-1 \)th coefficient of the characteristic polynomial of \( B_b|_n \) and the \( n-1 \)th coefficient of \( \Pi_{k=0}^n (x-u^k) \) converges to 0 for \( n\to\infty \).
Now we need a similar result for all the other coefficients...
As we want to show that the solutions of the sequence of characteristic polynomials converge to \( \{u^k: 0\le k\} \), maybe it suffices already to show that the coefficients of the sequence of characteristic polynomials converge to the coefficients of \( \Pi_{k=0}^n (x-u^k) \).
And indeed the \( n-1 \)th coefficient of \( \Pi_{k=0}^n (x-u^k) \) is \( \sum_{k=0}^n -u^k \), which is \( \frac{1}{1-u} \) in the limit.
So we have already one element of a proof, i.e.
the difference of the \( n-1 \)th coefficient of the characteristic polynomial of \( B_b|_n \) and the \( n-1 \)th coefficient of \( \Pi_{k=0}^n (x-u^k) \) converges to 0 for \( n\to\infty \).
Now we need a similar result for all the other coefficients...