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

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Concept |
Content |

Monotonicity of Limits
and Sandwich Theorem |
Taking limits respects weak inequalities and sandwiching a sequence by two converging sequences reveals its limit. The technique of sandwiching can be used to determine limits via nested intervals. |

Bolzano-Weierstrass
Theorem |
Every bounded sequence has at least one converging subsequence. |

Root Criterion |
An important criterion to prove absolute convergence by means of the behavior of the n-th roots of the underlying sequence's terms. |

Quotient Criterion |
An important criterion to prove absolute convergence by means of ratios of the underlying sequence's terms. |

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

Bounded Monotonic
Sequences |
If a sequence of real numbers is bounded and monotonic, then it is convergent. |