Convergent Sequence

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Definition (Convergent Sequence)

Let $(a_1,a_2,\ldots)$ be a sequence of real numbers.

• We say that $(a_n)$ converges to $L$ and write $(a_n)\to L$ if and only if for every $\varepsilon>0$, there exists a real number $M$ such that for every $n\in \mathbb{N}$ we have $n> M$ implies $|a_n-L|<\varepsilon$.
• When $(a_n)$ converges to $L$, we call $L$ the limit of $(a_n)$.
• We say that $(a_n)$ is a convergent sequence if and only if $(a_n)$ converges to $L$ for some real number $L$.
• We say that $(a_n)$ is a divergent sequence if and only if $(a_n)$ is not a convergent sequence.