Proof
We prove the existence and consider first the situation where
for all
for some fixed
. Then
-
is a polynomial of degree
which at the points
has value
. The polynomial
-
has at these points still a zero, but additionally at
its value is
. We denote this polynomial by
. Then
-
![{\displaystyle {}P=P_{1}+P_{2}+\cdots +P_{n}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20edc4baf66be140f949fe6399efa276a423051b)
is the polynomial looked for, because for the point
we have
-
![{\displaystyle {}P_{j}(a_{i})=0\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89e9117da13ff71c8cf6682733ecb480c77684b0)
for
and
.
The uniqueness follows from
fact.