Definition:Convergent Sequence/Note on Domain of N
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Definition
Let $\sequence {x_k}$ be a sequence.
- $\ds \lim_{n \mathop \to \infty} x_n \to l$
be the limit of $\sequence {x_k}$.
That is:
- $\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies \map d {x_n, l} < \epsilon$
Some sources insist that $N \in \N$ but this is not strictly necessary and can make proofs more cumbersome.
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.5$