Given the subset of $\mathbb{R}^{2}$ defined as follows:

$$S=\left\{(x, y) \in \mathbb{R}^{2} \mid y=\sin (1 / x), x>0\right\} \cup\{(0,0)\}$$

prove that its subset $S_{+}=\left\{(x, y) \in \mathbb{R}^{2} \mid y=\sin (1 / x), x>0\right\}$ is path-connected and that $S$ is connected, but not path-connected.