# Every point of Ext(C) lies on some tangent line to C

In our differential geometry class our professor gave the following problem as a homework problem:

Let $C$ be a smooth non-singular simple closed curve in plane. Prove that every point of the unbounded component of the complement $\mathbb{R}^2 \setminus C$ (Ext($C$)) lies on some tangent line to $C$.

The statement is simple but I have no idea how to proceed. Any hint is really appreciated.

Hint: For a point in the unbounded component, pick some random line through that point not intersecting your curve. Now move that line towards your curve $C$ until it touches it. At the moment it touches $C$ it will be tangent.
PS As far as I know, Ext(C) is not standard notation. I am guessing that you mean the unbounded component of $\mathbb{R}^2 \setminus C$.