Prove $\displaystyle \lim_{x \to 0}\frac{1}{x}$ does not exist.

Proof:

Asssume, to the contrary, that $\displaystyle \lim_{x \to 0}\frac{1}{x}$ exists. Then there exists a real number L such that $\displaystyle \lim_{x \to 0}\frac{1}{x}=L$. Let $\displaystyle \epsilon = 1$. Choose an integer $\displaystyle n$ such that $\displaystyle n>\lceil 1/\delta \rceil \geq 1$. Since $\displaystyle n>1/\epsilon$, it follows that $\displaystyle 0<1/n<\delta$. ...........

Question: In sequence $\displaystyle n$ denotes the term of the sequence. What does $\displaystyle n$ denote in this context?