I want to prove that the "infinite straight line" has Lebesgue measure zero as a subset of $\mathbb{R}^{2}$.
My approach is as follows. Let $\epsilon >0$ and let $A \subset \mathbb{R}^{2}$ denote the set of all points constituting the straight line. We can cover $A$ by a decreasing sequence of rectangles $B_{k}$ each of width $\frac{1}{n}$ and length $\frac{1}{n^{2k+1}}$, where $n \in \mathbb{N}$ is such that $\frac{1}{n} < \epsilon$ and $k \in \mathbb{Z}$. So the area of each rectangle is $\frac{1}{n^{2k}}$, and $A = \cup_{k=1}^{\infty} B_{k}$. Now $\frac{1}{n^{2k}} < \epsilon^{2k}$ and so the total area of the rectangles is
$\sum_{k=1}^{\infty} \frac{1}{n^{2k}} < \sum_{k=1}^{\infty} \epsilon^{2k} = \frac{\epsilon^{2}}{1 - \epsilon^{2}} < \epsilon$.
Thus we covered $A$ with a countable infinity of rectangles whose total area is less than $\epsilon$ and we are done.
Is this correct?