
$\{p_n\}$ is sequence of $X$ :
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 $p:\mathbb{N}\rightarrow X;n\mapsto p_n$

Sequence


Map preserve order $i:A\rightarrow A;k\mapsto i_k$\\$\forall k,k'\in A (k<k'\rightarrow i_{k}<i_{k'})$

Map preserve order


$p_{i_k}$ is sub sequence of $p_n$:
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$p':\mathbb{N}\rightarrow X;k\mapsto p_{i_k}$


$\{p_n\}$ converges to $p$:\\$\forall ε>0\exists n_0\in \mathbb{N}\forall n\in \mathbb{N}(n\geq n_0\rightarrow d(p_n,p)<ε)$
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$\{p_n\}$ is a cauchy sequence:
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$\forall ε>0\exists n_0\in \mathbb{N}\forall m,n\in\mathbb{N}(m,n\geq n_0\Rightarrow d(p_m,p_n)<ε)$

convergent sequence $\Rightarrow $ Cauchy sequence

$(X,d)$ is complete : all cauchy sequence of $X,d$ is convergent sequence