Consider the two tangent open discs $\displaystyle \{x^2+y^2<1\}$ and $\displaystyle \{(x-2)^2+y^2<1\}$ in $\displaystyle \mathbb{R}^2$.

How can show that the union of one disc and the closure of the other is a connected subset of $\displaystyle \mathbb{R}^2$?