For a given function we consider the following boundary value problem:$$-\dfrac{d^2u(x)}{dx^2} + u(x) = f(x)\text{ for }0<x<1,$$ with boundary conditions $$u(x = 0) = 0\text{ and }u(x = 1) = 0.$$ A finite difference method is used on a uniform mesh with $N$ intervals and mesh width $h = 1/N$.
Exercise: Suppose that $A^h$ is the coefficient matrix for the discretised equation. The eigenvalues of $A^h$ are given by $\lambda_k^h(A^h) = 1 + \dfrac{2}{h^2}2\sin^2\bigg(\dfrac{k\pi h}{2}\bigg)$. Compute the condition number of $A^h$ in the $2$-norm as a function of the mesh width $h$.
The given solution: The condition number is given by $$\kappa_2(A^h) = \dfrac{\lambda_\max(A^h)}{\lambda_\min(A^h)} = \dfrac{\lambda^{N-1}(A^h)}{\lambda^1(A^h)} = \dfrac{2 -2\cos\big((N-1)\pi h\big) + h^2}{2 - 2\cos\big(\pi h\big) + h^2}.$$
Question: Why is this correct? How do we know that $\lambda_\max(A^h) = 2 - 2\cos\big((N-1)\pi h\big) + h^2$? When the exercise says that the eigenvalues of $A^h$ are given by $\lambda_k^h(A^h) = 1 + \dfrac{2}{h^2}2\sin^2\bigg(\dfrac{k\pi h}{2}\bigg)$, what is meant with the subscript $k$?
Thanks in advance!