Problem:
Given a set of $n+1$ data points ($x_i, y_i$) where no two $x_i$ are the same, one is looking for a polynomial $p$ of degree at most $n$ with the property $p(x_i) = y_i$ for all $i∈ [0, n
]$.
Question:
What is the most intuitive way to prove that polynomial $p$ always exists? Of course, there are many ways to formally prove it, but I'm looking for a simple, informal explanation of it.