Taylor's polynomial, proof for existence and uniqueness

Unfortunately I'm not talking about Taylor's theorem, but Taylor's polynomial. The problem stated is : Let be a function that can be derived at least times in . Define with precision what is the Taylor's polynomial of order of in . Prove that it exists and that it is unique.

My attempt : I define it : The Taylor's polynomial of order of in is the only polynomial of degree lesser or equal to that satisfies , , with union . Now I have to prove that it exists and that it's unique. How can I prove that?

Looking to my notes, the proof is not done, but it says to write the polynomial this way : Let the polynomial be , with and then after a few algebra with the derivatives of the polynomial, it says . The funny part is that now it says, ) shows the existence while ) shows the uniqueness.

I'm very confused. If I understood well, I must show that **if** the polynomial can be written such as above, **then** and that **if** , **then** the polynomial can be written as above. But I'm not sure, so I'm asking you. And in case of an affirmation, would you help me? I thought about induction but realized it was not worth it.