We claim that
and prove it by induction.
First of all, let us observe that the sum is nonzero only for
, i.e. only for such k for which
. Such k exists only for
, i.e. for
. Hence, if we investigate the term
, we find out that
is the Kronecker delta.
Let us now proceed to the induction and consider
Further, let us assume, that the investigated equality is fullfilled for
and we show that it holds for
, as well. Thus, we start with
Now we apply the inductive step and write
The last sum vanishes for
, hence we may consider only
. In addition, if we take into account, that
, then there holds
Since for the last sum it holds
and employing the equality (*) completes the proof.