Are the convergent sequences dense in the bounded sequences? Since it would be comfortable for something I am currently trying to prove if this would hold I wanted to ask here whether it is true that $c$ is dense in $l^{\infty}(\mathbb{N})$?

Consider the sequence $(y_1,y_2,y_3,y_4,\cdots)=(0,1,0,1,\cdots)$. Suppose that there is a convergente sequence $x_n$ such that $$|x_n-y_n|\leq \epsilon \tag{1}$$ 
where $\epsilon$ is  small number. We conclude from $(1)$ that $x_{2n+1}$ is close to $0$ while $x_{2n}$ is close to $1$. This implies that $$|x_{2n+1}-x_{2n}|\ge 1+\delta$$
for some positive small $\delta$ depending on $\epsilon$.
