I am new to numerical analysis and this is the first thing I came across. It says on my textbook that interpolation polynomials are unique and to prove that it was assumed that let there be two such polynomials $\psi(x)$, $\phi(x)$ of degree n, given (n+1) values of the unknown function $f(x)$ are provided. I don't understand why this condition is important. Also since the value of the two polynomials at (n+1) points will be equal we have $F(x) = \psi(x) - \phi(x) = 0$ at those points, where $F(x)$ is at most a polynomial of degree n. Now it's said
$F(x)$, being at most an n degree polynomial, has (n+1) roots which is impossible unless it vanishes at all points, hence $\psi(x) = \phi(x)$
Why was it assumed that $\psi(x)$ & $\phi(x)$ have to be n degree polynomials in the first place? And why does $F(x)$ have (n+1) roots?
We suppose there are two polynomial $P_n(x)$ and $Q_n(x)$ such that for $i=0,1,\cdots,n$ $$P_n(x_i)=f(x_i)$$ and $$Q_n(x_i)=f(x_i)$$ let $$R_n(x)=P_n(x)-Q_n(x)$$ for $i=0,1,\cdots,n$ we have $$R_n(x_i)=P_n(x_i)-Q_n(x_i)=f(x_i)-f(x_i)=0$$ therefor $R_n(x_i)$ has $n+1$ roots whereas degree of $R_n$ is $n$, thus we can say $$R_n(x)=0$$ in the other words $$P_n(x)=Q_n(x)$$