I am working on a homework, where we have to proof that GMRES finds an exact solution of $Ax = b$ in at most m steps (with A being an $m \times m$-matrix). The proof is split up into several steps, and I have already solved all but one. The remaining one is:
Under the assumption in (c - see below), show that the solution x to the system of equations $Ax = b$ lies in $\mathcal{K}_n$. Conclude that GMRES has found the solution to $Ax = b$ in step n.
Here is part c (I already proved that one): Assume that at some n, $h_{n+1,n} = 0$ in the Arnoldi iteration (Arnoldi breakdown). Show that $\mathcal{K}_n$ is an invariant subspace of A, i.e. $Av \in \mathcal{K}_n$, for every $v \in \mathcal{K}_n$.
Note: The question here is such that the solution is already found in $n < m$ steps if there happens to be an Arnoldi breakdown, i.e., $h_{n+1,n} = 0$.
My idea so far was to say that if $x$ (the real solution) were not in $\mathcal{K}_n$, than there has to exist a vector $v \in \mathcal{K}_n$ such that $A^{k}v = \alpha x$ - i.e. subsequent Arnoldi iterations will transform $v$ into $x$ (apart from the length - that's why there is the factor $\alpha$). That would be proof by contradiction.
But I don't feel like my assumption that this vector exists is necessarily true. So any advice for me?
Thanks