Given a sequence of increasing real numbers $T = \{t_1 < t_2 < ... < t_{d+1}\}$, the $d+1$ Lagrange polynomials $L_i(t)$ of degree $d$ are defined as
$$L_i(t) = \prod_{\substack{1\leqslant j \leqslant d+1\\ j\neq i}} \frac{t-t_j}{t_i-t_j}.$$
From the definition it is clear that $L_i(t_j) = \delta_{ij}$.
I'm looking for different proofs showing that the Lagrange functions partition unity, i.e.
$$\sum_{1 \leqslant k \leqslant d+1} L_k(t) = 1 \quad \forall \, t.$$
I came up with the following (somewhat informal) proof. Due to the Kronecker delta property, the sum of the Lagrange polynomials of degree $d$ is equal to $1$ at the values $t_j \in T$. Since there are $d+1$ such values, and the sum is a degree $d$ polynomial (which is uniquely defined by $d+1$ values), the sum has to be equal to $1$ for all values of $t$. $\;\square$
I'd be very interested in alternative proofs (or references to them).