applicability: $\mathbb {R}$
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
