Can someone explain this proof, please?

Problem B1 (supposedly "easy" or even "trivial")
at the IMC (International Maths Competition) in 2001, in Prague, reads:

a_i and b_j are non-negative reals such that

(a_0 + a_1x + a_2x^2 + ... + a_{n-1}x^{n-1} + x^n)
(b_0 + b_1x + b_2x^2 + ... + b_{m-1}x^{m-1} + x^m)
= 1 + x + x^2 + ... + x^{m+n}.

Show that all a_i and b_j are 0 or 1.

The solution offered reads (in toto):

Solution. Multiply the left hand side polynomials. We obtain the
following
equalities:
a0 b0 = 1, a0 b1 + a1 b0 = 1, . . .
Among them one can find equations
a0 + a1 bs-1 + a2 bs-2 + . . . = 1
and
b0 + b1 ar-1 + b2 ar-2 + . . . = 1.
From these equations it follows that a0 , b0 1. Taking into account
that
a0 b0 = 1 we can see that a0 = b0 = 1.
Now looking at the following equations we notice that all a's must be
less
than or equal to 1. The same statement holds for the b's. It follows from
a0 b1 + a1 b0 = 1 that one of the numbers a1 , b1 equals 0 while the other
one must
be 1. Follow by induction.


I don't understand this proof
(possibly because I just arrived home 26 hours late).

I can see that one of a_1,b_1 is 0 and the other 1,
though not by the argument given.

But what is the induction over?

--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
.