# Definition:Pointwise Limit

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## Definition

Let $S$ be a set.

Let $\sequence {f_n}_{n \mathop \in \N}$, $f_n: S \to \R$ be a sequence of real-valued functions.

Suppose that for all $s \in S$, the limit:

- $\ds \lim_{n \mathop \to \infty} \map {f_n} s$

exists.

This definition needs to be completed.In particular: Incorporate infinite limitsYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding or completing the definition.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{DefinitionWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

Then the **pointwise limit of $\sequence {f_n}_{n \mathop \in \N}$**, denoted $\ds \lim_{n \mathop \to \infty} f_n: S \to \R$, is defined as:

- $\forall s \in S: \ds \map {\paren {\lim_{n \mathop \to \infty} f_n} } s := \lim_{n \mathop \to \infty} \map {f_n} s$

**Pointwise limit** thence is an instance of a pointwise operation on real-valued functions.

This definition needs to be completed.In particular: of course, makes sense in arbitrary metric/topological space. cover thisYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding or completing the definition.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{DefinitionWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |