Prove this is true (you should aim to prove mathematically t...

Prompt

Prove this is true (you should aim to prove mathematically that a solution EXISTS, not necessarily FINDING the solution itself). Do not stop until you have proved it is true: > > **Question Q:** *Is some unit fraction $\frac{1}{M}$ (where $M \ge 2$) a finite sum of masses of pairwise separated positive integer intervals of length $\ge 2$?* > > ### Breaking that down into plain terms: > 1. **A Unit Fraction:** A fraction with 1 in the numerator (e.g., $\frac{1}{42}$, $\frac{1}{1640}$, etc.). > 2. **Interval Mass:** The sum of the reciprocals of **strictly positive** integers in a continuous sequence of at least two numbers. For example, the mass of the interval $[4, 5]$ is $\frac{1}{4} + \frac{1}{5}$. **(Negative integers and 0 are strictly forbidden).** > 3. **Pairwise Separated:** The intervals used must have a gap of at least one integer between them (e.g., you can use $[4,5]$ and $[7,9]$, but not $[4,5]$ and $[6,7]$)