Sequences and Limits

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✓ 15 results · all verified · 7 also independently AI-judged

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1 · Prerequisites

Real Analysis

Set Theory

2 · Logical flowchart

3 · Items

Definition

Sequences and Limits

3 · Items

Sequences of reals: bounded, eventually, frequently, tails, subsequences

Convergence depends only on the tail

A sequence has at most one limit

Every convergent sequence is bounded

Every convergent sequence is Cauchy

A null sequence times a bounded sequence is null

Algebra of limits: sums, scalar multiples, products and quotients

Limits preserve non-strict inequalities

The squeeze theorem

The absolute value is compatible with limits

Divergence to +∞ and to -∞

For positive terms, null and divergence to +∞ are reciprocal

A strictly increasing index map satisfies nₖ ≥ k

Subsequences inherit the limit

Conventions for sequences: indexing, eventually, lim, and rational ε

FALSE: every bounded sequence converges

FALSE: limits preserve strict inequalities

FALSE: a convergent subsequence forces the sequence to converge