Injective functions between an interval and a square

Question

A task given to us is about proving that the interval $X=[0,1]$ and the square $Y=[0,1] × [0,1]$ are equipotent.

Part 1 asks to find an injective function $f : X \rightarrow Y$

Part 2 asks to find an injective function $g : Y \rightarrow X$

I can't seem to find the answer to these two and I imagine they are more obvious than what it seems. I know that the Cantor-Bernstein-Schröder theorem has to be used eventually. Any help would be greatly appreciated.

Answer 1

Hints: Part 1: You are missing something easy that I believe you know how to do.

Part 2: Consider the map $(.a_1 a_2 \dots, .b_1 b_2 \dots ) \to .a_1 b_1 a_2 b_2 \dots,$ where $.a_1 a_2 \dots $ etc. are decimal expansions. This map doesn't quite work, but tweaking it might.