# Proof

(related to Proposition: Cauchy Criterion)

Let $$s_j:=\sum_{k=0}^j x_k$$ be the $$j$$-th partial sum of the infinite real series $$\sum_{k=0}^\infty x_k$$ and let $$(s_j)_{j\in\mathbb N}$$ be the real 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 real sequence $$(s_j)_{j\in\mathbb N}$$ is a real Cauchy sequence. According to the completeness principle every real Cauchy sequence is convergent. If follows that $$\sum_{k=0}^\infty x_k$$ is a convergent real series.

Github: ### References

#### Bibliography

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