Problem: Three nonnegative real numbers $r_1$, $r_2$, $r_3$ are written on a blackboard.
These numbers have the property
that there exist integers $a_1$, $a_2$, $a_3$, not all zero,
satisfying $a_1r_1 + a_2r_2 + a_3r_3 = 0$.
We are permitted to perform the following operation:
find two numbers $x$, $y$ on the blackboard with $x \le y$,
then erase $y$ and write $y - x$ in its place.
Prove that after a finite number of such operations,
we can end up with at least one $0$ on the blackboard.

Solution: We first show we can decrease the quantity
$\left\lvert a_1 \right\rvert + \left\lvert a_2 \right\rvert + \left\lvert a_3 \right\rvert$
as long as $0 \notin \left\{ a_1,a_2,a_3 \right\}$.
Assume $a_1 > 0$ and $r_1 > r_2 > r_3$
without loss of generality and consider two cases.

\begin{itemize}
  \ii Suppose $a_2 > 0$ or $a_3 > 0$; these cases are identical.
  (One cannot have both $a_2 > 0$ and $a_3 > 0$.)
  If $a_2 > 0$ then $a_3 < 0$ and get
  \[ 0 = a_1r_1 + a_2r_2 + a_3r_3 > a_1r_3 + a_3r_3 \implies a_1 + a_3 < 0 \]
  so $\left\lvert a_1 + a_3 \right\rvert < \left\lvert a_3 \right\rvert$,
  and hence we perform $(r_1,r_2,r_3) \mapsto (r_1-r_3,r_2,r_3)$.

  \ii Both $a_2 < 0$ and $a_3 < 0$.
  Assume for contradiction that $\left\lvert a_1+a_2 \right\rvert \ge -a_2$
  and $\left\lvert a_1+a_3 \right\rvert \ge -a_3$ both hold
  (if either fails then we use $(r_1,r_2,r_3) \mapsto (r_1-r_2,r_2,r_3)$
  and $(r_1,r_2,r_3) \mapsto (r_1-r_3,r_2,r_3)$, respectively).
  Clearly $a_1+a_2$ and $a_1+a_3$ are both positive in this case,
  so we get $a_1 + 2a_2$ and $a_1 + 2a_3 \ge 0$; adding gives $a_1+a_2+a_3 \ge 0$.
  But
  \begin{align*}
    0 &= a_1r_1 + a_2r_2 + a_3r_3 \\
    &> a_1r_2 + a_2r_2 + a_3r_2 \\
    &= r_2(a_1+a_2+a_3)  \\
    \implies 0 &< a_1+a_2+a_3.
  \end{align*}
\end{itemize}

Since this covers all cases, we see that we can always decrease
$\left\lvert a_1 \right\rvert + \left\lvert a_2 \right\rvert + \left\lvert a_3 \right\rvert$
whenever $0 \notin \left\{ a_1,a_2,a_3 \right\}$.
Because the $a_i$ are integers this cannot occur indefinitely,
so eventually one of the $a_i$'s is zero.
At this point we can just apply the Euclidean Algorithm, so we're done.