# indirect proof of identity theorem of power series

 $\displaystyle\sum_{n=0}^{\infty}a_{n}(z-z_{0})^{n}\;=\;\sum_{n=0}^{\infty}b_{n% }(z-z_{0})^{n}$ (1)

is valid in the set of points $z$ presumed in the theorem (http://planetmath.org/IdentityTheoremOfPowerSeries) to be proved.

Antithesis:  There are integers $n$ such that  $a_{n}\neq b_{n}$;  let $\nu$ ($\geqq 0$) be least of them.

We can choose from the point set an infinite sequence$z_{1},\,z_{2},\,z_{3},\,\ldots$  which converges to $z_{0}$ with  $z_{n}\neq z_{0}$  for every $n$.  Let $z$ in the equation (1) belong to  $\{z_{1},\,z_{2},\,z_{3},\,\ldots\}$  and let’s divide both of (1) by $(z-z_{0})^{\nu}$ which is distinct from zero; we then have

 $\displaystyle\underbrace{a_{\nu}+a_{\nu+1}(z-z_{0})+a_{\nu+2}(z-z_{0})^{2}+% \ldots}_{f(z)}\,=\,\underbrace{b_{\nu}+b_{\nu+1}(z-z_{0})+b_{\nu+2}(z-z_{0})^{% 2}+\ldots}_{g(z)}$ (2)

Let here $z$ to tend $z_{0}$ along the points $z_{1},\,z_{2},\,z_{3},\,\ldots$, i.e. we take the limits $\lim_{n\to\infty}f(z_{n})$ and $\lim_{n\to\infty}g(z_{n})$.  Because the sum of power series is always a continuous function, we see that in (2),

 $\mathrm{left\,side\,}\longrightarrow f(z_{0})=a_{\nu}\quad\mathrm{and}\quad% \mathrm{right\,side\,}\longrightarrow g(z_{0})=b_{\nu}$

But all the time, the left and of (2) are equal, and thus also the limits.  So we must have  $a_{\nu}=b_{\nu}$,  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 IndirectProofOfIdentityTheoremOfPowerSeries 2013-03-22 16:47:48 2013-03-22 16:47:48 pahio (2872) pahio (2872) 9 pahio (2872) Proof msc 40A30 msc 30B10