Given n points, show there is a unique polynomial through all of them

Question

if x₀, x₁,..., x_n are distinct points in [a,b] and $A$ = { a₀, a₁,..., a_n } ∈ Rⁿ⁺¹, show there is a unique polynomial $p$_A of degree at most "n" such that $p$_A(x_j) = a_j for each "j".

Then, show that there is a constant $M$ such that for all "a", ||$p$_a||_∞ ≤ $M$||$A$||₂.

In the second part, ||.||_∞ is the "sup norm" or "max norm" on the space of continuous functions $C$([a,b]).