Convergence of the sequence

• categories: Real Analysis, Definition

Definition : convergence of the Sequence

$\{p_n\}$ converges in $X$ (metric space) if $\exists p\in X\mid \forall ϵ\exists N\mid \forall n\ge N⟹dist(p,p_n)<ϵ$ ${\lim }_{n\to \infty }{p}_n=p$

Properties

• Unique
• Sequence is bounded
• If $p$ is a limit point ⇒ there is a sequence
• If countable infinite number of distinct elements ⇒ Limit point
• $\iff$ for every neighborhood infinitely many points inside
• lim if sum = sum of limits
• multiplying by constant is allowed
• multiplying of sequences is allowed
• taking the subsequence keeps the limit

Theorems :

1

Bounded monotonic sequence converges