# Proof

Let $$s_j:=\sum_{k=0}^j x_k$$ be the $$j$$-th partial sum of the infinite complex series $$\sum_{k=0}^\infty x_k$$ and let $$(s_j)_{j\in\mathbb N}$$ be the complex sequence of these partial sums. The proposed condition for the convergence states that for every $$\epsilon > 0$$ there is an index $$N(\epsilon)\in\mathbb N$$ such that

$\left|\sum_{k=m}^n x_k\right| < \epsilon\quad\quad \text{for all}\quad n\ge m\ge N(\epsilon).$

Since

$s_n-s_{m-1}=\sum_{k=m}^n x_k,$

we get

$\left|s_n-s_{m-1}\right| < \epsilon\quad\quad \text{for all}\quad n\ge m\ge N(\epsilon).$

This means that the complex sequence $$(s_j)_{j\in\mathbb N}$$ is a complex Cauchy sequence. According to the completeness principle for complex numbers, every complex Cauchy sequence is convergent. If follows that $$\sum_{k=0}^\infty x_k$$ is a convergent complex series.

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### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer VerĂ¤nderlichen", Vieweg Studium, 1983