Say I have a sequence of positive numbers $(a_n)_{n\in \mathbb{N}}$. I know that the sum of all numbers is bounded by some constant, i.e. it holds $$\sum_{n=1}^\infty a_n < C$$ for some constant $C$. Now I inspect $\sum_{n=k}^\infty a_n$ and let $k$ grow to infinity. My intuition it tells me that $$\lim_{k\rightarrow \infty}\sum_{n=k}^\infty a_n$$ must be zero because $\sum_{n=k}^\infty a_n$ is being "squashed" towards $C$ and cannot exceed $C$, but I cannot show this formally. Can you please give me some hints?
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2Does this answer your question? If $a_n \geq 0$ $\forall n \in \mathbb{N}$ and $\sum_{n=1}^ \infty a_n < \infty$, what can be said about $\lim_{k\to\infty}\sum_{n=k}^\infty a_n$? – Martin R May 25 '23 at 17:38
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Also: https://math.stackexchange.com/q/2210075/42969 – Martin R May 25 '23 at 17:38
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Yes thank you and sorry for the duplicate, I didn't see the other discussions. – Duck71 May 25 '23 at 18:38