Product of Uniformly Convergent Sequences of Bounded Functions is Uniformly Convergent
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This article, or a section of it, needs explaining, namely:
for the product to be defined, we probably want $Y$ to have some more structure
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This theorem requires a proof..
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 It has been suggested that this page or section be merged into Uniformly Convergent Sequence Multiplied with Function. (Discuss) |
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 $S$.
for the product to be defined, we probably want $Y$ to have some more structure
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Proof
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