Let $X_0, X_1, X_2, \dotsc$ be sets such that $X_0 \neq \emptyset$ and
$$Card(X_0) < Card(X_1) < Card(X_2) < \dotsc$$
Prove that $Card(\bigcup\limits_{i=0}^{\infty}X_i) < Card(\prod\limits_{i=0}^{\infty}X_i)$.
My guess is to show that there exists a function $f: \bigcup\limits_{i=0}^{\infty}X_i \rightarrow \prod\limits_{i=0}^{\infty}X_i$ that is injective and show that it cannot be surjective. It seems pretty clear to me though I do not know where to begin with this proof.