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Title: Sequence of Functions
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Series: Real Analysis
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Chapter: Continuous Functions
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YouTube-Title: Real Analysis 23 | Sequence of Functions
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Bright video: https://youtu.be/RM2hytsyMpc
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Dark video: https://youtu.be/199yh1SWw5o
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Ad-free video: Watch Vimeo video
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Exercise Download PDF sheets
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: ra23_sub_eng.srt missing
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Download bright video: Link on Vimeo
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Download dark video: Link on Vimeo
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Timestamps
00:00 Intro
00:19 Definition function
01:06 Continuous function
02:20 Definition bounded function
03:36 Definition sequence of functions
05:57 Credits
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Subtitle in English (n/a)
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Quiz Content
Q1: What is not a correct definition for a bounded function $f: I \rightarrow \mathbb{R}$?
A1: $\mathrm{Ran}(f)$ is bounded.
A2: $f[I]$ is bounded.
A3: ${ x \mid f(x) \in \mathbb{R} }$ is bounded.
A4: ${ f(x) \mid x \in I }$ is bounded.
A5: $\displaystyle \sup_{x \in I} |f(x)| < \infty$.
A6: $\displaystyle \sup_{x \in I} f(x) < \infty$ and $\displaystyle \inf_{x \in I} f(x) > -\infty$.
Q2: What is a sequence of functions?
A1: Just another name for a sequence.
A2: A sequence, where the sequence members are functions $f_n: I \rightarrow \mathbb{R}$.
A3: A map $\mathbb{N} \rightarrow \mathbb{N}$.
Q3: If $x \in I$ is fixed and $(f_1, f_2, f_3, \ldots)$ is a sequence of functions $f_n: I \rightarrow \mathbb{R}$, then:
A1: $(f_1(x), f_2(x), f_3(x), \ldots)$ is an ordinary sequence of real numbers.
A2: $(f_1(x), f_2(x), f_3(x), \ldots)$ is always a convergent sequence.
A3: $(f_1(x), f_2(x), f_3(x), \ldots)$ is also a sequence of functions.
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Last update: 2025-01
