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{\displaystyle {\begin{aligned}d\vdash \ \mathbb {X} \times \mathbb {Y} \subseteq \mathbb {R} ^{2}\ \ \land \ \mathbb {X} \times \mathbb {Y} \neq \varnothing \ \ \land \ \ \mathrm {S_{f}} =\{\mathrm {f} |\ \ \mathrm {f} \in {\mathcal {P}}(\mathbb {X} \times \mathbb {Y} )\ \land \ \mathrm {f} :\mathbb {X} \mapsto \mathbb {Y} \}\ \ \land \ \mathrm {u} :\mathbb {N} \mapsto \mathrm {S_{f}} \\\ \land \quad \mathrm {v} \in \mathrm {S_{f}} \quad \to \quad (\mathrm {[The\ sequence]\ u\ converges\ uniformly\ to\ v.} \ \leftrightarrow \ \lim _{n\to \infty }\mathrm {u} _{n}=\mathrm {v} )\end{aligned}}}
Примечание
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{\displaystyle ~\mathbb {X} \times \mathbb {Y} }
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{\displaystyle ~\mathbb {X} }
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{\displaystyle ~\mathbb {Y} }
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{\displaystyle ~{\mathcal {P}}(\mathbb {X} \times \mathbb {Y} )}
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{\displaystyle ~\mathbb {X} \times \mathbb {Y} }
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{\displaystyle ~\mathrm {f} :\mathbb {X} \mapsto \mathbb {Y} }
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{\displaystyle ~\mathbb {X} }
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{\displaystyle ~\mathbb {Y} }
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{\displaystyle ~\mathrm {S_{f}} }
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{\displaystyle ~\mathbb {X} }
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{\displaystyle ~\mathbb {Y} }
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{\displaystyle ~\mathrm {u} :\mathbb {N} \mapsto \mathrm {S_{f}} \ \Leftrightarrow \ \mathrm {u} =\{\langle n,\mathrm {u} _{n}\rangle |\ \ \langle n,\mathrm {u} _{n}\rangle \in \mathbb {N} \times \mathrm {S_{f}} \ \ \land \ \ \mathrm {u} _{n}=\{\langle x,y\rangle |\ \ \langle x,y\rangle \in \mathbb {X} \times \mathbb {Y} \ \ \land \ \ y=u_{n}(x)\}\ \}}
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{\displaystyle ~\mathrm {v} \in \mathrm {S_{f}} \ \Leftrightarrow \ \mathrm {v} =\{\langle x,y\rangle |\ \ \langle x,y\rangle \in \mathbb {X} \times \mathbb {Y} \ \ \land \ \ y=v(x)\}}
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{\displaystyle ~\lim _{n\to \infty }\mathrm {u} _{n}=\mathrm {v} \ \Leftrightarrow \ \forall _{\varepsilon \ \in \ (0,\infty )}\ \exists _{N\ \in \ \mathbb {N} }\ \forall _{x\ \in \ \mathbb {X} }\ \forall _{n\ \in \ \mathbb {N} }\ (n>N\ \to \ |u_{n}(x)-v(x)|<\varepsilon )}
Разные формы записи утверждения "[The sequence]
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{\displaystyle ~\mathrm {u} :\mathbb {N} \mapsto \mathrm {S_{f}} }
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{\displaystyle ~\lim _{n\to \infty }\mathrm {u} _{n}=\mathrm {v} \quad \Leftrightarrow \quad \lim _{n\to \infty }\ \sup _{x\in X}\ |u_{n}(x)-v(x)|=0\quad \Leftrightarrow \quad \mathrm {u} _{n}\rightrightarrows \mathrm {v} }
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{\displaystyle ~t\vdash \ \lim _{n\to \infty }\mathrm {u} _{n}=\mathrm {v} \ \to \ \forall _{x\ \in \ \mathbb {X} }\ (\lim _{n\to \infty }u_{n}(x)=v(x)\ )}
![{\displaystyle {\begin{aligned}d\vdash \ \mathbb {X} \times \mathbb {Y} \subseteq \mathbb {R} ^{2}\ \ \land \ \mathbb {X} \times \mathbb {Y} \neq \varnothing \ \ \land \ \ \mathrm {S_{f}} =\{\mathrm {f} |\ \ \mathrm {f} \in {\mathcal {P}}(\mathbb {X} \times \mathbb {Y} )\ \land \ \mathrm {f} :\mathbb {X} \mapsto \mathbb {Y} \}\ \ \land \ \mathrm {u} :\mathbb {N} \mapsto \mathrm {S_{f}} \\\ \land \quad \mathrm {v} \in \mathrm {S_{f}} \quad \to \quad (\mathrm {[The\ sequence]\ u\ converges\ uniformly\ to\ v.} \ \leftrightarrow \ \lim _{n\to \infty }\mathrm {u} _{n}=\mathrm {v} )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/daf6b30461328ac7bdfbfb12fe50b612db09350a)
