Socle of socle of module is the socle of the

Step 1
I write SM for the soccel of M. It is clear that ${\displaystyle SSM\subset SM}$. Now take ${\displaystyle x\in SM}$. The element x may be written as a finite linear combination ${\displaystyle x=y_1+\cdots +y_n}$ where each ${\displaystyle y_r}$ is contained in a simple submodule ${\displaystyle V_r}$ of M. But each ${\displaystyle V_r}$ is also a simple submodule of SM, so in particular ${\displaystyle y_r}$ is contained in SSM. Since SSM is a submodule, it follows that x is contained in it.