Prove all polynomials $p\in V$ is written of the form: $p=\sum_{i=0}^n L_ip(a_i)$ where $L_i(x)=\prod_{j\not=i}\frac{x-a_j}{a_i-a_j}$ and $a_i\in\mathbb{R}$ with $1\leq i\leq n$.
My work:
for each $i$ consider $L_i(x)=\prod_{j\not=i}\frac{x-a_j}{a_i-a_j}$
If $x=a_j$ with $j\not=i$ then $L_i(x)=\prod_{j\not=i}\frac{x-a_j}{a_i-a_j}=0$
Moreover,
If $x=a_j$ with $j=i$ then $L_i(x)=\prod_{j\not=i}\frac{x-a_j}{a_i-a_j}=1$
Then,
$\sum_{i=0}^n L_i(a_j)p(a_i)=\sum_{i=0}^n \delta_{ij}p(a_i)=p(a_j)$
In consequence,
$p=\sum_{i=0}^n L_ip(a_i)$
Is good the proof? Is convicent?