Let this "line" be the $x$ axis, and let the heads of the $1390$ ants be labeled $A_1, A_2,..., A_{1390}$. Now let the coordinates of $A_n=(x_n,y_n)$ and WLOG, let $x_i\le x_j$ if $i\le j$. Since the heads of $2$ ants are always larger than $2\text{cm}$, the circles with radius $1$ and centers $A_i$ must be disjoint. Also, since the ants can't deviate at least $1\text{cm}$ away from the line, all the circles will be contained in the rectangle with height $2+1+1=4$ and width $x_{1390}-x_1+2$. Since the circles are disjoint and are contained in the rectangle mentioned above, we can make the following inequality using area: \[1390\pi\le 4(x_{1390}-x_1+2)\] \[x_{1390}-x_{1}\ge \frac{1390\pi}4-2>\frac{1390(3)}4-2=1042.5-2>1000\text{cm}=10\text{m} \] Thus $x_{1390}-x_{1}>10\text{m}$. By the Pythagorean Theorem, $A_{1390}A_1=\sqrt{(x_{1390}-x_{1})^2+(y_{1390}-y{1})^2}\geq x_{1390}-x_{1}>10\text{m}$. Therefore we have shown $A_{1390}A_1$ is greater than $10\text{m}$. $\blacksquare$