Definiton. A sequence ${a_{n}}$ of real numbers is said to convergence to the number $a$, or have limit equal to $a$, if, for each $\varepsilon >0$, there is a 'real number' $N$ such that $\mid a_{n} -a\mid$ < $\varepsilon$ whenever $n > N$.
My question is that why we take $N$ as a real number? It should be natural number, shouldn't it?
If $N\in\mathbb{N}$ is valid then any $N^*\in\mathbb{R}$ larger than $N$ will be valid, because if $n>N^*$ then $n>N$, assumed that $n\in\mathbb{N}$.