How to formally prove that a line segment $[z,w]$ can be covered by overlapping disks? Let's say that $[z,w]$, i.e. the closed line segement between $z,w\in \mathbb{C}$ is of length less than $R$. How can I argue for the fact that the line segment can be covered by $5$ disks of radius $R/10$ with centres in $[z,w)$ (halfopen segment)?
 A: We have $[z, w] = \{(1-\lambda)z + \lambda w : \lambda \in [0,1]\}$. Consider the disks centered at:
$$\left(1 - \frac{1}{10}\right)z + \frac1{10}w$$
$$\left(1 - \frac{3}{10}\right)z + \frac3{10}w$$
$$\left(1 - \frac{5}{10}\right)z + \frac5{10}w$$
$$\left(1 - \frac{7}{10}\right)z + \frac7{10}w$$
$$\left(1 - \frac{9}{10}\right)z + \frac9{10}w$$
Those disks cover $[z,w]$. Indeed, take an arbitrary $(1-\lambda)z + \lambda w$ for some $\lambda \in [0,1]$.
There exists $i \in \{0, 1, \ldots,9\}$ such that $$\frac{i}{10} \le \lambda \le \frac{i+1}{10}$$
Without loss of generality assume that $i$ is odd. Consider the disk centered at $\left(1 - \frac{i}{10}\right)z + \frac{i}{10}w$.
We have:
\begin{align}\left\|(1-\lambda)z + \lambda w - \left(1-\frac{i}{10}\right)z -
 \frac{i}{10} w\right\|&= \left\|\left(\frac{i}{10} -
 \lambda\right)z + \left(\lambda - \frac{i}{10}\right)w\right\|\\
&=\underbrace{\left(\lambda - \frac{i}{10}\right)}_{\le\frac{1}{10}}\underbrace{\|z-w\|}_{<R}\\
&< \frac{R}{10}
\end{align}
Therefore, $(1-\lambda)z + \lambda w$ is contained in the disk around $\left(1 - \frac{i}{10}\right)z + \frac{i}{10}w$.
If $i$ were even, we would consider the disk centered at $\left(1 - \frac{i+1}{10}\right)z + \frac{i+1}{10}w$.
We conclude that the disks cover $[z,w]$.
