First, is $P_n$ $n$ or rather $n+1$ dimensional real vector space of polynomials of degree at most $n$ ? Here, $n$ indicates that it should be $n$ but the basis $1,x,x^2,x^3,...,x^n$ has $n+1$ elements, which could indicate that the dimension is after all $n+1$. In the above I was thinking as follows: by some trivial case $1$ could be generated from the other powers.
Finally, I understand that if $1,x,x^2,x^3,...,x^n$ form a base $P_n$ then I would like to also see that the fundamental polynomials $$l_i(x)=\frac{(x-x_0)...(x-x_{i-1})(x-x_{i+1})...(x-x_n)}{(x_i-x_0)...(x_i-x_{i-1})(x_i-x_{i+1})...(x_i-x_n)}$$ also form a base. What are (almost) all different ways to see this fact ?
You know that the $n+1$ polynomials $L_i$ satisfy $$ L_i(x_j) = \delta_{ij}. $$ It follows that the only linear combination that can vanish at all the $x_j$ is the combination with all $0$ coefficients. Therefore the $L_i$ are independent. Since there are $n+1$ of them in a space of that dimension they form a basis.