(related to Proposition: Monotony Criterion)
Let \(\sum_{k=0}^\infty x_k\) be an infinite series with non-negative terms \(x_n\ge 0\) for all \(n\in\mathbb N\) and let sequence \((s_n)_{n\in\mathbb N}\) of its partial sums \(s_n:=\sum_{k=0}^n x_k\) be bounded. Then the sequence \((s_n)_{n\in\mathbb N}\) is monotonically increasing, and it follows from the monotone convergence theorem that \((s_n)_{n\in\mathbb N}\) is convergent.
Let the infinite series be convergent. Then the sequence \((s_n)_{n\in\mathbb N}\) of its partial sums is convergent. It follows from the corresponding proposition that convergent sequences are bounded.
