This is a question from the Understanding Analysis book:
Informally speaking, the sequence $\sqrt{n}$ "converges to infinity". Imitate the logical structure of Definition 2.2.3 to create a rigorous definition for the mathematical statement $\lim x_n = \infty$. Use this definition to prove $\lim \sqrt{n} = \infty$.
Definition 2.2.3:
A sequence $(a_n)$ converges to a real number $a$ if, for every positive number $\epsilon$, there exists an $N \in \mathbb{N}$ such that whenever $n \geq N$ it follows that $|a_n - a| < \epsilon$.
I don't know what it means for a definition to be "rigorous". Would the following definition suffice?
A sequence $(a_n)$ "converges to infinity" if, for every positive number $\epsilon$, there exists an $N \in \mathbb{N}$ such that whenever $n \geq N$, it follows that $a_n > \epsilon$.
Not really sure if the above definition would work though. Would appreciate help if it's not a rigorous definition.