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tukeywilliams · #35 June 30th, 2007, 12:06 AM

just wanted to point out: isn't a sequence a function defined as $f: \mathbb{Z^{+}} \rightarrow A$ which means a sequence in the set $A$ ? So the codomain doesn't have to be $\mathbb{R}$ but can be an arbitrary set $A$ ? Actually $\mathbb{N}$ and $\mathbb{Z^{+}}$ are equivalent right? And a sequence is null when $\lim_{n \rightarrow \infty} f(n) = 0$ when $\forall \epsilon \in \mathbb{R^{+}}, \exists N \in \mathbb{Z^{+}}, \forall n \in \mathbb{Z^{+}} (n \geq N \Rightarrow |f(n)| < \epsilon)$ . Also, I like to think of functions as follows: pretend you have a box subdivided into smaller boxes. These smaller boxes represent the elements of the codomain, and the points in the boxes represent the elements of the domain. So a function orders the points in the smaller boxes.