The concept of sequences but for functions instead of real numbers.

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Concept |
Content |

Bounded Sets,
Maxima and Minima |
The values inside a set of real numbers can be bounded. |

Supremum and
Infimum of Sets |
Bounded sets always have an supremum and infimum which are generalizations of maximum and minimum. |

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. |

Image and
Preimage |
Via images and preimages we describe how functions work on sets. |

Sequences |
These object are needed to define limits later on. |

Sets |
Sets are the basic building blocks for a lot of mathematics. In order to rigorously define numbers and doing real analysis, we need to know how to work with sets. |

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**Definition 1** (Bounded functions). *Let \(I\) be a set. Then we call a function \(f:I\rightarrow \mathbb{K}\)
**bounded*, if \[\sup\{|f(x)|\,:\,x\in
I\}<\infty.\]

**Definition 2** (Sequence of functions). *Let \(I\) be a set and for all \(n \in \mathbb{N}\), let \(f_n :I\rightarrow \mathbb{K}\) be
**bounded*. Then \((f_1, f_2, f_3, f_4,
\dots)\) is a sequence of functions.

Not that, for any fixed point \(x \in
I\), we get an ordinary sequence of real numbers \[\begin{aligned}
(f_1(x), f_2(x), f_3(x), f_4(x), \dots).
\end{aligned}\]

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