Cover the plane with strips

 In the 2D plane, a strip is the space between (and including) two parallel lines (which extend to infinity in both directions). The width of a strip is the distance between the two parallel lines that make up a strip. The width of a strip is non-zero (otherwise we just call it a line).

Can you fully cover the 2D plane with strips the sums of whose widths is finite? (Proof required of course).

Scroll down for a solution.

Consider a circle of radius $R$. A strip of width $w_i$ can cover an area at most $2w_iR$.

So strips of total width $w$ (infinite number of strips or not), can cover at most an area of $2wR$.

For sufficiently large $R$, $\pi R^2 \gt 2wR$. So we cannot cover the plane with strips the sum of widths of which is finite.