When studying the general angular momentum $\textbf{J}$, which is defined as a vector operator with its components being Hermitian operators satisfying the commutation relations
\begin{align*} \textbf{J} \times \textbf{J} = i\hbar \textbf{J} , \end{align*} and the simultaneous eigenvectors of $\textbf{J}^2$ and an abitrary component $J_z$, my book stated that the eigenvalues $j$ and $m$ are either integer (including zero) or half-odd-integer values. They came to this conclusion by using the raising and lowering operators which are defined as \begin{align*} J_{\pm} = J_x \pm iJ_y. \end{align*} More specifically, because of the inequality \begin{align*} j(j+1) \geq m^2 \end{align*} there must be an upper and lower bound $m_T , m_B$ which differ from each other an integer amount $n$.
However, what ensures that these eigenvalues are the only possible values? How can it not be that the spectrum of eigenvalues is continuous? Say for example we forget about the raising and lowering operators and just postulate that there exists a simultaneous eigenvector of both $\textbf{J}^2$ and $J_z$ with $j = 2.3$ and $m = 2.3$. Is there a way to contradict this assumption?
I have already found an answer using the representation theory of Lie algebras, but since my mathematical background is limited, I would like to have a simpler explanation. If this is the only formal explanation, then I would like to know that and move on :). The book I use for studying QM is Bransden & Joachain (chapter 6.5).
