Lightwing wrote: (a)Given any natural number N, prove that there exists a strictly increasing sequence of N positive integers in harmonic progression. (b)Prove that there cannot exist a strictly increasing infinite sequence of positive integers which is in harmonic progression. a) Choose for exemple $x_k=\frac{N(N+1)!}{N+1-k}$ b) If the two first elements are $m,n$ with $0<m<n$, the $k^{th}$ element is $\frac {mn}{n-k(n-m)}$ And when $k\to+\infty$, this number becomes negative.