Criticize the following "proof" that $\mathbb S^2$ is simply connected: Let $f$ be a loop in $\mathbb S^2$ based at $x_0$. Choose a point $p$ of $\mathbb S^2$ not lying in the image of $f$. Since $\mathbb S^2 -p$ is homeomorphic with $\mathbb R^2$ and $\mathbb R^2$ is simply connected, the loop $f$ is path homotopic to the constant loop.
So when the author puts the word proof inside ”...” it means that given is not a proof but a fallacy. But I couldn't find anything wrong with this. Can somebody help me with this?