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set {s=\sum\frac{1}{x_i}|x_i\in\mathbb{N},s<d}

Source: 16-th Hungary-Israel Binational Mathematical Competition 2003

March 30, 2007
inductionnumber theory proposednumber theory

Problem Statement

Let d>0d > 0 be an arbitrary real number. Consider the set Sn(d)={s=1x1+1x2+...+1xn∣xi∈N,s<d}S_{n}(d)=\{s=\frac{1}{x_{1}}+\frac{1}{x_{2}}+...+\frac{1}{x_{n}}|x_{i}\in\mathbb{N},s<d\}. Prove that Sn(d)S_{n}(d) has a maximum element.
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