 Solution

The sphere $S^2$ is path-connected. For $u,v\in S^2$ which are not an antipodal pair, a continuous path isThe denominator never vanishes: a zero would make $u$ and $v$ antipodal. If $v=-u$, choose a unit vector $w$ perpendicular to $u$ and concatenate the paths from $u$ to $w$ and from $w$ to $v$ of the same form. Thus $S^2$ is connected. The quotient map $q_S$ is continuous and surjective, so the continuous image of a connected space theorem proves that ${\mathbb{RP}}^2$ is connected. In fact composing the paths above with $q_S$ also proves path connectedness.