Proof with rationals and irrationals

[major_maths]

Homework Statement

Show that any rational in the interval (0,1] can be expressed as a finite sum r=1/q₁+1/q₂+...+1/q_n where the q_j are integers and q₁<q₂<...<q_n.

Homework Equations

The Attempt at a Solution

Let x\inQ and 0<x\leq1.
Prove \existsq₁, q₂, ..., q_n\inN with q₁<q₂<...<q_n.

My professor suggests using the greedy algorithm but I don't understand how that would help the proof.

 

[Hurkyl]

Since you're having trouble tackling the problem for all rationals, have you tried first working on the simpler problem of just considering some rationals? Maybe certain classes of them, or just pick seven at random and see what you can do?