• Title: Cauchy Sequences

  • Series: Start Learning Reals

  • Parent Series: Start Learning Mathematics

  • Chapter: Real Numbers

  • YouTube-Title: Start Learning Reals 1 | Cauchy Sequences

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  • Quiz Content

    Q1: What is not correct for the absolute value in $\mathbb{Q}$?

    A1: $| x | \geq 0$ for all $x \in \mathbb{Q}$

    A2: $| x y | = | x | |y|$ for all $x,y \in \mathbb{Q}$

    A3: $| x + y | = | x | + |y|$ for all $x,y \in \mathbb{Q}$

    A4: $|x| = 0 $ if and only if $x = 0$

    Q2: What is the correct formulation for a sequence in $\mathbb{Q}$?

    A1: It’s a map $\mathbb{N} \rightarrow \mathbb{Q}$.

    A2: It’s a map $\mathbb{Q} \rightarrow \mathbb{N}$.

    A3: It’s a map $\mathbb{Q} \rightarrow \mathbb{Q}$.

    A4: It’s a map $\mathbb{N} \rightarrow \mathbb{N}$.

    Q3: What is the short formulation for a Cauchy sequence $(x_n)_{n \in \mathbb{N}}$ in $\mathbb{Q}$?

    A1: $\forall \varepsilon > 0 $ $ ~~ \exists N \in \mathbb{N} $ $ ~~ \forall n,m \geq N ~:~$ $ |x_n - x_m| < \varepsilon$

    A2: $\exists \varepsilon > 0 $ $ ~~ \exists N \in \mathbb{N} $ $ ~~ \forall n,m \leq N ~:~$ $ |x_n - x_m| < \varepsilon$

    A3: $\exists \varepsilon > 0 $ $ ~~ \forall N \in \mathbb{N} $ $ ~~ \forall n,m \leq N ~:~$ $ |x_n - x_m| < \varepsilon$

    A4: $\exists \varepsilon > 0 $ $ ~~ \forall N \in \mathbb{N} $ $ ~~ \exists n,m \leq N ~:~$ $ |x_n - x_m| < \varepsilon$

    A5: $\forall \varepsilon > 0 $ $ ~~ \exists N \in \mathbb{N} $ $ ~~ \forall n,m \geq N ~:~$ $ |x_n - x_m| > \varepsilon$

    A6: $\forall \varepsilon > 0 $ $ ~~ \forall N \in \mathbb{N} $ $ ~~ \exists n,m \leq N ~:~$ $ |x_n - x_m| < \varepsilon$

  • Last update: 2024-10

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