Definition:Convergent Sequence/Rational Numbers

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Let $\sequence {x_k}$ be a sequence in $\Q$.

$\sequence {x_k}$ converges to the limit $l \in \R$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies \size {x_n - l} < \epsilon$

where $\size x$ is the absolute value of $x$.


The definition of convergence of a sequence of rational numbers is equivalent to the definition of convergence of a real sequence.

Note on Domain of $N$

Some sources insist that $N \in \N$ but this is not strictly necessary and can make proofs more cumbersome.