I just need help understanding this problem:

Suppose that \{X_i,\Omega_i) : i\in I\} is a family of topological spaces and suppose that \bold{x} and \bold{y} are members of \prod \{X_i : i\in I\}. If \bold{x} and \bold{y} differ in only one coordinate, prove that they must lie in the same connected component.

What we have to work with:

Let (X,\Omega) and (Y,\Theta) be topological spaces. If (X,\Omega) is connected and the function f: X \to Y is continuous relative to \Omega and \Theta, then f(X) is connected as a subspace of (Y,\Theta).

Suppose that \{X_i,\Omega_i) : i\in I\} is a family of topological spaces. Fix i\in I; and for all j\in I - \{i\}, fix a_j \in X_j. Define the function f_i: X_i \to \prod\{X_i : i\in I\} by f_i(x) = \bold{x}, where \pi_i(\bold{x}) = x and \pi_j(\bold{x}) = a_j when j\neq i. Using this map, it can be shown that (X_i,\Omega_i) is homeomorphic to the subspace f_i(X_i).

Both of these have been proven in class. I'm just failing to see how to use these when one coordinate differs. Any help is greatly appreciated!