In one article about Hopf algebras I found a statement that if $C$ is a coalgebra and $(C_n)_{n\ge 1}$ is a coalgebra filtration of $C$ ( i.e. filtration of $C$ as a vector space, such that $\Delta(C_n)\subset \sum\limits_{i=0}^{n}C_{i}\otimes C_{n-i}$ ) then $C_1$ contains $C_0=\mathrm{Corad}(C)$ - coradical of $C$.
How to prove it ? I need only hints.
This is not true for arbitrary coalgebra filtrations. For example, we can always set $C=C_0=C_i$, for all $i$, then $\Delta(C_i)=\Delta(C)\subset C\otimes C=\sum_i C_i\otimes C_{n-i}$.
Is it possible that the paper talks about the coradical filtration? For it, we have $C_0=Corad(C)$ by definition.