I am reading C. L . Liu "Introduction to Combinatorial Mathematics".
I think his proof of the following theorem is not correct.
Am I right?
p.312 Theorem 12-1
If a linear programming problem has a feasible solution, it also has a basic feasible solution.
His proof:
Suppose that $\hat{x_1}, \hat{x_2}, \dots, \hat{x_{r+m}}$ is a feasible solution. If all of $\hat{x_1}, \hat{x_2}, \dots, \hat{x_{r+m}}$ are zero, they constitute a basic feasible solution. (In fact, this is a degenerate case in which the feasible region consists only of this one point.) $\cdots$
For example, I think $(x_1, x_2, x_3, x_4) = (0, 0, 0, 0)$ and $(x_1, x_2, x_3, x_4) = (1, 0, 2, 1)$ are feasible solutions of the following linear programming problem:
$$
-2 x_1 + x_2 + x_3 = 0 \\
-x_1 + x_2 + x_4 = 0 \\
x_1, x_2, x_3, x_4 \ge 0
$$