Prove that the real projective line cannot be embedded into Euclidean space

Well, $\mathbb{R}P^1$ is a subspace of $\mathbb{R}P^2$ (namely, any projective line in the projective plane), so since $\mathbb{R}P^2$ embeds in $\mathbb{R}^4$, so does $\mathbb{R}P^1$. But actually, you can do better: $\mathbb{R}P^1$ is just a circle, so it embeds in $\mathbb{R}^2$. Indeed, if you remove a point from a circle, the resulting space is homeomorphic to $\mathbb{R}$, so a circle is the 1-point compactification of $\mathbb{R}$ and hence homeomorphic to $\mathbb{R}P^1$.

In lieu of a thousand words, here's an embedding of the real projective line (the set of lines in the plane through the origin) in the Cartesian plane:

$\mathbb{R}P^1$ is just the circle $S^1$ and this cannot be embedded in the reals: an infinite connected subset of $\mathbb{R}$ (is an interval so) always has a cut-point (a point we can remove to leave a disconnected subset), and the circle remains connected if we remove any point.