I ask about proof of Courant's nodal domain theorem.
Let $M$ be a Riemannian manifold. Let $0\le\lambda_1 \le \lambda_2 \le \cdots$ be eigenvalues of $M$, and $\{\phi_1,\phi_2,\cdots$} be a complete orthonormal basis of $L^2(M)$ such that $\phi_j$ is an eigenfunction of $\lambda_j$ for each $j=1,2,\cdots$.
Summary of Proof:
Let $V_1,V_2,\cdots,V_k,V_{k+1},\cdots$ be nodal domains of $\phi_k$. For each $i=1,2,\cdots,k$,
$$\Psi_i= \begin{cases} \phi_k &\text{on $V_i$}& \\ 0 &\text{on $\overline{M}$ -$V_i$}& \end{cases}$$ Then there exists a nontrivial function $$f=\sum_{i=1}^k c_i\Psi_i$$ satisfying $0=(f,\phi_1)=\cdots=(f,\phi_{k-1})$.
※(,) is inner product of $L^2(M)$.
Then $f$ is an eigenfunction of $\lambda_k$ by Rayleigh's theorem (or Max-Min theoorem) and Green's formula.And $f$ is vanishing identically on $V_{k+1}$.
By the maximum principle, $f$ is vanishing on $M$.This is contradiction.
Question: I want to know how to apply the maximum principle to $f$.
In particular, whether $f$ satisfies $\Delta f\ge0 (\mbox{ or } \le0)$ or not?
I'd appreciate if you could answer this questions.