A note on connected cofiltered coalgebras, conilpotent coalgebras and Hopf algebras

This notes gives a proof that a connected coaugmented cofiltered coalgebra is a conilpotent coalgebra and thus a connected coaugmented cofiltered bialgebra is a Hopf algebra. This applies in particular to a connected coaugmented cograded coalgebra and a connected coaugmented cograded bialgebra.