Let $\lambda$ be an eigenvalue of $B$ (the square matrix of order $q$ with coefficient $(i,j)$ equal to 1 if $|i-j|=1$ and 0 otherwise) and let $Y = \left(\begin{array}{c} y_{1} \\ \vdots \\ y_{q} \end{array}\right)$ be an associated eigenvector. By considering a coefficient of $Y$ whose absolute value is maximal, show that $\lambda \in [-2, 2]$ and justify the existence of an element $\theta$ of $[0, \pi]$, such that $\lambda = 2\cos\theta$.