Definitions:
Connected: Not separated
Separated: If $C$ is a subset of a metric space, then $(A, B)$ is a separation of $C$ if $C = A \cup B$, $A \neq \varnothing$, $B \neq \varnothing$, and we cannot have that $\{x_n\} \in A$, $x_n \to x$, with $x \in B$ (and also vice versa).
We know that an interval (in $\mathbb{R}$) is connected. So knowing this, I tried looking up a proof that the product of connected sets is connected, but the proofs I read use the notion of homeomorphism which we aren't allowed to use (since we haven't learned it). What would be the direct method of showing $\mathbb{R^2}$ is connected using only our sequential definition of connectedness?