Convergence in a sequence space For any $n\in \mathbb{N}$ consider the following sequence: 
$$a_n=\left(\underbrace{\frac{1}{n}, \ldots, \frac{1}{n}}_{n \text{ elements}}, 0, 0, 0, 0, \ldots \right).$$
Is there a way to introduce a norm on a sequence space in which $a_n$ will converge to the zero sequence? 

Sure, for instance you can consider ($l_{\infty}, ||\cdot ||_{\infty}$), which is the space of bounded sequences with the supremum norm. 
