\begin{array}{llclccclcl} \text{maximize} & c_1x_1 & + & c_2x_2 & + & \dots & + & c_nx_n \\\text{subject to} & a_{11}x_1 & + & a_{12}x_2 & + & \dots & + & a_{1n}x_n & \le & 0\\ & a_{21}x_1 & + & a_{22}x_2 & + & \dots & + & a_{2n}x_n & \le & 0\\ & & \vdots & & \vdots & & \vdots & & \vdots & \\ & a_{m1}x_1 & + & a_{m2}x_2 & + & \dots & + & a_{mn}x_n & \le & 0\\ & & & & & x_1, & \dots, & x_n & \ge & 0 \\ \end{array}
Show that either $(x_1,\dots,x_n) = (0,\dots,0)$ is the optimal solution or the LP is unbounded.
If $x=0$ is not optimal, that means that there is an $x \geq 0$ for which $c^Tx > 0$ and $Ax \leq 0$. Then $\alpha x$ ($\alpha \geq 0$) is also feasible, and by $\alpha \to \infty$ the objective can be made arbitrarily large.