applicability: $\mathbb {R}$

# Definition: Convergent Real Sequence

A convergent real sequence is a real sequence $$(x_n)_{n\in\mathbb N}$$, which is convergent in the metric space of real numbers $$(\mathbb R,|~|)$$. In other words, $$(x_n)_{n\in\mathbb N}$$ is convergent to the number $$x\in\mathbb R$$, i.e. for each $$\epsilon > 0$$ there exists an $$N\in\mathbb N$$ with $| x_n-x | < \epsilon\quad\quad \text{ for all }n\ge N.$

If $$(x_n)_{n\in\mathbb N}$$ is convergent to the number $$x\in\mathbb R$$, we write $\lim_{n\to\infty} x_n=x.$

In this case, the convergent sequence is also said to be convergent to the number (or tending to the number) $x$.

Applications: 1
Chapters: 2 3
Corollaries: 4 5 6
Definitions: 7 8 9 10 11 12 13 14 15
Examples: 16
Lemmas: 17 18
Proofs: 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58
Propositions: 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83
Sections: 84
Solutions: 85
Theorems: 86 87 88 89

Github: ### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
2. Forster Otto: "Analysis 2, Differentialrechnung im $$\mathbb R^n$$, Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984