Coradical filtration and socle series $C_n=\mathrm{Soc}^{n+1}(C)$

Question

I am reading the book Hopf Algebras and Their Actions on Rings by Susan Montgomery. In page 64, she said $C_n=\mathrm{Soc}^{n+1}(C)$, where $C$ is a coalgebra with coradical filtration $\{C_n \}$ and the socle series is defined inductively by $\mathrm{Soc}^{n+1}(C)=\mathrm{Soc} (C/\mathrm{Soc}^n(C))$ viewing $C$ as a left $C^*$-module.

I know $\mathrm{Soc}(C)=C_0$ which is $5.1.8$ in her book. But how can I prove the case for $n \geq 1$?

Thanks in advance.