A congruence for a double harmonic sum

Resumen

In this short note, our primary purpose is to prove the congruence

\[\sum_{k=1}^{\frac{p-1}{2}} \frac{(-1)^k}{k} \sum_{j=\lfloor\frac{k}{2}\rfloor+1}^{k} \frac{1}{2j-1} \equiv 0 \mod p\]

Along the way, a number of auxiliary results of independent interest are found.