Suppose that for $a_n\geq b_n$ for all $n$. Show that $\varliminf_{n \to \infty} a_n\geq \varliminf_{n \to \infty} b_n$.

Question

This is what I have so far:

Since $a_n\ge b_n$ for every $n$ then we have that $\inf\{a_n; n\ge k\} \ge \inf\{b_n; n\ge k\}$ for every $n$. When we take the limit as $n\rightarrow \infty$ we get $$\varliminf_{n \to \infty} a_n\geq \varliminf_{n \to \infty} b_n$$

Am I on the right track?

Answer 1

Yes, this solution is correct.