Statement:
Let $X$ and $Y$ be sets. If $Y$ is countable and there exists an injective function from $X$ to $Y$, then $X$ is countable.
My proof:
Suppose $Y$ is countable and there exists an injective $f: X \mapsto Y$. Since $Y$ is countable, there exists an injective function $g: Y \mapsto \mathbb{Z}^+$. Let $x_1, x_2 \in X$ and suppose $f(g(x_1)) = f(g(x_2))$. Since $f$ is is injective, $g(x_1) = g(x_2)$. Since $g$ is injective $x_1 = x_2$, so $f \circ g$ is injective. This means that $X$ is countable.