These object are needed to define limits later on.

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__Show requirements__

Concept |
Content |

Real Numbers |
In a real analysis, the real numbers are the largest number set we need. They satisfy axioms that represent the idea of a number line. |

Maps |
Maps are the mathematical formulation of a machine that gets inputs and generate outputs. On both sides, sets are needed for the domain and the codomain. |

Natural Numbers
and Induction |
Using natural numbers is our first mathematical abstraction as children. Mathematical induction is an important technique of proof. |

__Show consequences__

Concept |
Content |

Bounded
Sequences |
Sequences can be bounded from above and from below. The values of a bounded sequence cannot become arbitrarily large. |

Cauchy
Sequences |
The sequence members of a Cauchy Sequence eventually become arbitrarily close to each other. |

Convergence |
Convergent sequences have a well-defined limit. |

Series and
Partial Sums |
A series is a sequence of partial sums. |

Sequences of
Bounded Functions |
The concept of sequences but for functions instead of real numbers. |

Limits of Functions |
How function evaluations change when the argument approaches a certain point. |

**Definition 1**. *Let \(M\) be a set. A **sequence in \(M\)* is a map \(a:\mathbb{N}\to M\) or \(a:\mathbb{N}_0 \to M\).

We use the following symbols for sequences: \[(a_n)_{n\in\mathbb{N}},\qquad (a_n),\qquad
(a_n)_{n=1}^\infty,\quad (a_1,a_2,a_3,\ldots).\]

**Example 3**.

\(a_n=(-1)^n\), then \((a_n)_{n\in\mathbb{N}}=((-1)^n)_{n\in\mathbb{N}}=(-1,1,-1,1,-1,1,\ldots)\);

\(a_n=\frac1n\), then \((a_n)_{n\in\mathbb{N}}=(\frac1n)_{n\in\mathbb{N}}=(1,\frac12,\frac13,\frac14,\frac15,\frac16,\ldots)\);

\(a_n=\mathrm{i}^n\) (\(\mathrm{i}\) is the imaginary unit), then
\((a_n)_{n\in\mathbb{N}}=(\mathrm{i}^n)_{n\in\mathbb{N}}=(\mathrm{i},-1,-\mathrm{i},1,\mathrm{i},-1,\ldots)\);

\(a_n=\frac1{2^n}\), then \((a_n)_{n\in\mathbb{N}}=(\frac1{2^n})_{n\in\mathbb{N}}=(1,\frac12,\frac14,\frac18,\frac1{16},\frac1{32},\ldots)\);

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