A Natural Limit Definition

Often, the first exposure one gets to rigorous mathematics is the definition of a limit. Let’s consider what this is for a sequence. We say \lim_{n\rightarrow \infty} a_n = A  if

\displaystyle  \forall \epsilon \in \mathbb{R}^+\quad\exists N \text{  s.t.}\quad\quad n > N \implies |a_n-A| \leq \epsilon   

This, at first sight, is ugly. It takes a while to even understand what it’s saying, longer to see why it works, and much longer to apply it. It’s intimidating to say the least. I feel, however, there is another version that makes the idea of limits simple and very natural giving a deep insight into what a limit really is.

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