Show that $HK^+$ is a coideal where $K$ is a subHopfalgebra of $H$, $K^+=\mathrm{ker}(\epsilon)\cap K$.
Because $\epsilon(ha)=\epsilon(h)\epsilon(a)=0$ for $a\in K^+$, $\forall h \in H$. What we need to prove is $$ \Delta(HK^+)\subset HK^+ \otimes H + H \otimes HK^+ $$
I proved a special case for $H=kG$, $K=<g,g'>$, $g\not= g' \in G$. Here, $K^+=\{\lambda(g-g')| \lambda \in k\}$. Note that $$ \Delta(g-g')=\Delta(g)-\Delta(g')= g\otimes g+ g'\otimes g'. $$ For any $h\in H$, $$ \begin{aligned} \Delta(h(g-g')) = &h_1g\otimes h_2g- h_1g'\otimes h_2g'\\ = &h_1g\otimes h_2g- h_1g'\otimes h_2g\\ &+h_1g'\otimes h_2g- h_1g'\otimes h_2g'\\ = &h_1(g-g')\otimes h_2g+h_1g'\otimes h_2(g-g')\\ \in &HK^+ \otimes H + H \otimes HK^+, \end{aligned} $$ since $\epsilon(g-g')=\epsilon(g)-\epsilon(g')=0$.
But how to prove the general case?
I solved this problem on my own. It is obvious if one notice that $$ \Delta(a)=\sum_{i=1}^n a_i \otimes a_i'= \sum_{i=1}^{n-1} \{ (a_i -\frac{\epsilon(a_i)}{\epsilon(a_n)} a_n) \otimes a_i' + \frac{\epsilon(a_i)}{\epsilon(a_n)} [a_n \otimes (a_i'-\frac{\epsilon(a_i')}{\epsilon(a_n')} a_n')] \} $$