$\left|\prod_{j = 1}^{\infty} F\right| = |F|$ if $F$ is a no finite field?

Question

Let $F$ be a no finite field. How can I show the equality $$ \left|\prod_{j = 1}^{\infty} F\right| = |F|? $$ The inequality $|F| \leq \left|\prod_{j = 1}^{\infty} F\right|$ is clear, but how can I show that $\left|\prod_{j = 1}^{\infty} F\right| \leq |F|$? Must I differenciate the two cases $|F| = |\mathbb{R}|$ and $|F| = |\mathbb{Q}|$? Is there no possibility to get $\left|\prod_{j = 1}^{\infty} F\right| = |\mathbb{R}|$ and $|F| = |\mathbb{Q}|$?