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Suppose m,n > N. Then f_m(x) = f_n(x) = -1 when x \in [0, \frac{1}{2}-\frac{1}{N}]. Similarly, f_m(x) = f_n(x) = +1 when x \in [\frac{1}{2}+\frac{1}{N},1]. And |f_m(x)-f_n(x)| \leq 1 ...
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You are trying to show that: (P^{-1}AP-\lambda_{1}I)^{k_{1}}\dots(P^{-1}AP-\lambda_{m}I)^{k_{m}}=0 let's take one subspace: (P^{-1}AP-\lambda_{1}I)^{k_{1}} then it's equal to \begin{bmatrix} ...
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