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# 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 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 sides 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 right side 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).

## Mathematics Subject Classification

40A30*no label found*30B10

*no label found*

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