Here is the problem:
$$L = \sum_{i\in [1,n]} a_i x_i \\ s.t. \sum_{i\in [1,n]} x_i = 1, \text{and } x_i >= 0 $$
I think this is a linear programming problem, and I tried to solve it by Lagrange multiplier but no idea at all.
However, I know that the solution is $$x = [0,...,x_k=1,0,0,0], \text{where } k = \text{argmax}_i(a_i)$$ , but how to solve it formally?
Let $x' = (x_1', \ldots, x_n')$ be the optimal solution (it exists since this is a classical linear programming problem with nonempty domain) and $a_k$ is the max coefficient in $L$. We have
$$ L(x') = \sum\limits_{i=1}^n a_ix_i' \leq \sum\limits_{i=1}^n a_kx_i' = a_k\sum\limits_{i=1}^nx_i' = a_k = L(x^*), $$ where $x^* = (0, \ldots, 0, x_k = 1, 0, \ldots, 0)$.
