 Solution

The spectral expansion givesFor $j\le n-2$, the coefficient tends to zero because $|{\lambda }_j|<|{\lambda }_n|$. Thus every possible limit belongs toThe identical argument with the coefficients $c_j$ provesThe statement is conditional because, when ${\lambda }_{n-1}=-{\lambda }_n$, the component along $w_{n-1}$ generally alternates rather than converges.