Proof of $ \limsup\limits_{n\rightarrow\infty} a_n b_n \le (\limsup\limits_{n\rightarrow\infty} a_n)(\limsup\limits_{n\rightarrow\infty} b_n)$

Question

$\{a_n\}$ and $\{b_n\}$ are sequences of positive real numbers. Here is my current plan.

Using $c \limsup\limits_{n\rightarrow\infty} a_n = \limsup\limits_{n\rightarrow\infty} ca_n $ where c is a real number. Taking $a = \limsup\limits_{n\rightarrow\infty} a_n$ right hand side of inequality becomes $\limsup\limits_{n\rightarrow\infty} ab_n$ which is greater than LHS.

Answer 1

Hint: Using the definition of the supremum, show (or perhaps simply note) that $$ \sup_{k\geq n} a_kb_k \leq \left(\sup_{k\geq n} a_k\right) \left(\sup_{k\geq n} b_k\right) $$