Let $X=\{x|x=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}, n\in \mathbb{N}\}$. Find $X \cap \mathbb{N}$ I am out of hints here. Its trivial to show $1$ is a solution. How to show it's the only solution?

Can someone please give me some hints?

Please do not use congruence, limits or derivatives because they are out of scope of the chapter which has this question.
The harmonic number is never an integer for $n&gt;1$. In fact (as shown in the article), the largest power of $2$ dividing the denominator in the simplest form, is the largest power of $2$ not exceeding $n$
The harmonic number is never an integer for $n&gt;1$. In fact (as shown in the article), the largest power of $2$ dividing the denominator in the simplest form, is the largest power of $2$ not exceeding $n$