I am given the fllowing problem:
Find the solution of the eigenvalue problem $$ y'' + \lambda y =0$$ with $y(0)=y(1)=0$
Let $h=\frac{1}{N}$. The FD discretization for the problem writes $$y_{n+1} - 2 y_n + y_{n-1} + \lambda_h h^2 y_n = 0 \quad (\star)$$ with $y_0=y_N=0$.
Since the components of the eigenvectors are the values of the eigenfunctions in nodes $x = nh$, study the convergence of the eigenvalues $\lambda_h$ and estimate the error.
My attempt:
First of all $y(x)=\sin(k \pi x)$, with $k \in \mathbb{Z}$. Now, from the last italic sentence, I know that $$y_n = y(x_n)=y(nh) = \sin(k \pi n h)$$
Therefore, plugging these relation for $y_n$ in $(\star)$, I find:
$$\lambda_h = \frac{2}{h^2} \bigl ( 1-\cos(k \pi h) \bigr)$$
Now, honestly, I don't know how to answer the question: it's clear that since the numerator is bounded by $2$, as $h \rightarrow 0$ the $\lambda_h$ diverges. But I don't know if I can stop here.