We are collecting donations to buy a new chair. We receive $ m$ donations in total $ d_1, d_2, …, d_m \in \mathbb{N}$ ($ m \geq 1$ and every donation $ d_i$ is whole-numbered, i.e $ \in \mathbb{N}$ ). A chair costs $ c$ dollars, where $ c \in [m].$ (Notation: $ [m]$ $ :=$ $ \{1, 2, …, m\}$ ).

Prove using mathematical induction over $ m$ , that there exists two numbers $ k, l$ with $ k \leq l$ such that the sum of the amount of donations $ \sum _{s=k} ^l {d_s}$ is exactly sufficient to purchase $ x$ chairs ($ x \in \mathbb{N}$ ) without any money being left.

I’m unable to find the correct approach to solve this. I’m sure the Pigeonhole Principle will be useful here, but I don’t know how to correctly apply it within the induction proof. Can someone point me in the right direction?