Product of path connected spaces is path connected

Will product of path connected topological spaces be necessarily path connected? Why or why not?

Give me some hints. Thank you.

This is very straightforward from applying the definitions involved: Assume $X=\prod_{i\in I}X_i$ with $X_i$ path-connected. Let $x=(x_i)_{i\in I}$, $y=(y_i)_{i\in I}$ be two points in $X$. By assumption, thee exist continuous paths $\gamma_i\colon[0,1]\to X_i$ with $\gamma_i(0)=x_i$ and $\gamma_i(1)=y_i$. By definition of product, there exists a unique continuous $\gamma\colon[0,1]\to X$ such that $\pi_i\circ\gamma=\gamma_i$ for all $i\in I$ where $\pi_i$ is the projection $X\to X_i$. That makes $\gamma$ a path from $x$ to $y$.

HINT: To get from $\langle x_0,y_0\rangle\in X\times Y$ to $\langle x_1,y_1\rangle\in X\times Y$, you can go from $\langle x_0,y_0\rangle$ to $\langle x_1,y_0\rangle$ to $\langle x_1,y_1\rangle$.