Let $\tau:c_0\rightarrow\ell_\infty$ be the inclusion map. Let $x_0=\{1,1,\cdots\}$. Then $x_0\in\ell_\infty-c_0$. Since $\tau(x)=x\in c_0$ for all $x\in c_0$, there is no $x\in c_0$ such that $\tau(x)=x_0$. Hence, $\tau:c_0\rightarrow\ell_\infty$ is not surjective.
But my professor told me my logic is wrong on this proof. Can any one help me on this? Thank you!
Your proof is perfectly correct. Perhaps your professor expects you to show more details in some steps, such as when you claim that $x_0\in\ell_\infty-c_0$. You could also spell out more explicitly why there is no $x\in c_0$ such that $\tau(x)=x_0$ (namely, that $\tau(x)=x$ so this would mean $x_0=x\in c_0$ but $x_0\not\in c_0$).