Let and be real power series which converge for . If = for , then prove that for all
From the givens, I conclude that , for all .
Next, if we assume (contrary to what is the case), that is not true for all indices , then there would have to exist a specific smallest index , such that . Since, therefore, for all indices smaller than we have , and thus , it follows that
, for all .
From it follows for the continuous[!] function that .
But this means that , and therefore , contrary to our assumption that is the smallest summation index for which that is not the case. Thus no such smallest index can exist, and this means that for all indices , we have , what was to be shown.