indirect proof of identity theorem of power series

n=0an(z-z0)n=n=0bn(z-z0)n (1)

is valid in the set of points z presumed in the theorem ( to be proved.

Antithesis:  There are integers n such that  anbn;  let ν (0) be least of them.

We can choose from the point set an infinite sequencez1,z2,z3,  which converges to z0 with  znz0  for every n.  Let z in the equation (1) belong to  {z1,z2,z3,}  and let’s divide both of (1) by (z-z0)ν which is distinct from zero; we then have

aν+aν+1(z-z0)+aν+2(z-z0)2+f(z)=bν+bν+1(z-z0)+bν+2(z-z0)2+g(z) (2)

Let here z to tend z0 along the points z1,z2,z3,, i.e. we take the limits limnf(zn) and limng(zn).  Because the sum of power series is always a continuous functionMathworldPlanetmath, we see that in (2),


But all the time, the left and of (2) are equal, and thus also the limits.  So we must have  aν=bν,  contrary to the antithesis.  We conclude that the antithesis is wrong.  This settles the proof.

Note.  I learned this proof from my venerable teacher, the number-theorist Kustaa Inkeri (1908–1997).

Title indirect proof of identity theorem of power series
Canonical name IndirectProofOfIdentityTheoremOfPowerSeries
Date of creation 2013-03-22 16:47:48
Last modified on 2013-03-22 16:47:48
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Proof
Classification msc 40A30
Classification msc 30B10