# Complicated problem involving mathematical induction

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?