Ive got this situation We have the final zero sequences C_00 and we have to prove that is dense in C_0.
The idea is proving that the clausure of C_00 is C_0.
And the C_00 is defined by the sequences In a field, such that exists a Natural N With X_n =0 for n greater than N.
Fix $x\in c_0$ and an arbitrary number $\delta >0.$ By the definition of the convergence there exists $n_0,$ such that $|x_n|<\delta$ for $n>n_0.$ Let $y$ be the sequence defined by $y_n=x_n$ for $1\le n\le n_0$ and $y_n=0$ for $n>n_0.$ Then $y\in c_{00}$ and $$\max_n|x_n-y_n| =\max_{n>n_0}|x_n|<\delta.$$