Suppose this process is endless. There exist real number $N<1$ such that all of $100$ initial numbers are smaller than $N$. Since if $a,b<N$, we get $\frac{a+\sqrt{a^2-4b}}{2}<N$. So, all numbers on the board will always smaller than $N$. Denote by $S$ and $P$ the sum and product of all numbers on the board. Let $S_0$ and $P_0$ be that of the initial $100$ numbers. Each move gives us $S\rightarrow S-b$ and $P\rightarrow \frac{P}{a}>\frac{P}{N}$. After $M$ moves, we get that $S<S_0$ and $P>\frac{P_0}{N^M}$. By A.M.-G.M., we get $S\geq 100\sqrt[100]{P}$. Not hard to see that this gives contradiction for sufficiently large $M$, done.