Define the terms connected and path connected for a topological space. If a topological space $X$ is path connected, prove that it is connected.

Consider the following subsets of $\mathbb{R}^{2}$ :

$$\begin{gathered} I=\{(x, 0): 0 \leq x \leq 1\}, \quad A=\left\{(0, y): \frac{1}{2} \leq y \leq 1\right\}, \text { and } \\ J_{n}=\left\{\left(n^{-1}, y\right): 0 \leq y \leq 1\right\} \quad \text { for } n \geq 1 \end{gathered}$$

Let

$$X=A \cup I \cup \bigcup_{n \geq 1} J_{n}$$

with the subspace (metric) topology. Prove that $X$ is connected.

[You may assume that any interval in $\mathbb{R}$ (with the usual topology) is connected.]