Numerical Methods: Euler to Runge–Kutta · Superchallenge problem

The eigenvalue problem $-y'' = \lambda y$ on $[0, 1]$ with $y(0) = y(1) = 0$ has smallest eigenvalue $\pi^2$. Discretise it on the grid $x_j = \dfrac{j}{N}$ ($h = \dfrac1N$) by replacing $-y''(x_j)$ with $\dfrac{-y_{j+1} + 2y_j - y_{j-1}}{h^2}$ for $j = 1, \ldots, N - 1$, with $y_0 = y_N = 0$. This is an $(N - 1) \times (N - 1)$ matrix eigenvalue problem; let $\Lambda_N$ be its smallest eigenvalue. Find $\displaystyle\lim_{N \to \infty} N^2\left(\pi^2 - \Lambda_N\right)$.