# Product of Uniformly Convergent Sequences of Bounded Functions is Uniformly Convergent

Jump to navigation
Jump to search

It has been suggested that this page or section be merged into Uniformly Convergent Sequence Multiplied with Function.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 `{{Mergeto}}` from the code. |

## Theorem

Let $X = \struct {A, d}$ and $Y = \struct {B, \rho}$ be metric spaces.

Let $\sequence {f_n}$ and $\sequence {g_n}$ be sequences of mappings from $X$ to $Y$.

Let $\sequence {f_n}$ and $\sequence {g_n}$ be uniformly convergent on some subspace $S$ of $X$.

$\forall n \in \N$, let $f_n$ and $g_n$ be bounded.

Then the sequence $\sequence {f_n g_n}$ is uniformly convergent on $S$.

This article, or a section of it, needs explaining.In particular: for the product to be defined, we probably want $Y$ to have some more structureYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.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 `{{Explain}}` from the code. |

## Proof

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.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 `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |