I would like to solve the following infinite linear system subject to $x_i \ge 0$ that minimizes $x_3$.

The third column contains no additional nonzero values beyond what is shown. Though the first 7 rows are jumbled, the pattern of entries is more clear for rows 8 and beyond.

One solution to the system is

$$\mathbf{x}= \left( 1,0,1,0,1,0,1,0,1,0,1,0,\dots \right)$$

But I would like to show that there is no solution where $x_3<1$. I know that infinite matrices are generally a devilish topic, but I'm hoping that standard linear programming and the simplex method (which I am not too familiar with) would get the job done for this matrix.