Prove the energy eigenstates are degenerate

[Philethan]

Homework Statement

Two observables $A_{1}$ and $A_{2}$ which do not involve time explicitly, are known not to commute, $[A_{1},A_{2}]\neq0,$
yet we also know that $A_{1}$ and $A_{2}$ both commute with the Hamiltonian: $[A_{1},H]=0\text{, }[A_{2},H]=0.$
Prove that the energy eigenstates are, in general, degenerate. Are there exceptions? As an example, you may think of the central-force problem $H=\textbf{p}^{2}/2m+V(r)$, with $A_{1}\rightarrow L_{z}$, $A_{2}\rightarrow L_{x}$.

Homework Equations

$[A_{1},A_{2}]\neq0,$
$[A_{1},H]=0\text{, }[A_{2},H]=0.$

The Attempt at a Solution

Please read my attached file. I type in latex. I really don't understand why I'm incorrect.

Thanks in advance!

 

[kuruman]

What do you know about the eigenstates of two operators that commute?
What do you know about the eigenstates of two operators that do not commute?