quantum mechanics - Question about angular momentum operator

To show that the eigenvalue to $L^2$ is proportional to $\hbar^2$ is shown from

$L_z=xP_y-yP_x$

$p_y=-i\hbar\frac{\partial}{\partial y}$

$p_x=-i\hbar\frac{\partial}{\partial x}$

$L_z=-i\hbar(x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x})$

$L_z=\hbar^2(x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x})^2$

In the same way can $L_x^2$ and $L_y^2$ be obtained and since

$L^2=L_x^2+L_y^2+L_z^2$ (I) we see that $L^2$ is proportional to $\hbar^2$

So we can establish this eigenvalue equation

$L^2|\alpha,\beta>=\hbar^2\alpha|\alpha, \beta>$

From solving this it is obtained that:

$\alpha=l(l+1)$

My problem is that in the derivations that I have found they use the defintion of Raisng and lowering operators:

$L_+=Lx+iLy$ Raising

$L_-=Lx-iLy$ Lowering

They increase and decrease respectively $\alpha$ with 1 integer value. But why can we know that the square of the angular momentum eigenvalue can be increased or decreased in inger values? The rest of the derivation makes sense to me. But I need one of two explanations:

1: Prior to the ladder operators are there a way to show that $\alpha$ can only have integer values?

or 2: A way to derive the ladder operators and why they are how they are and deriving why they work on $\alpha$ the undefined part of the the square of the angualar momentum eigenvalue to give solutions to $\alpha$

Here is an example of a derivation of the type I am refering to

http://en.wikipedia.org/wiki/Angular_momentum_operator#Derivation_using_ladder_operators