1) If an ≥ 0 and bn ≥ 0, prove that lim sup anbn ≤ (lim sup an)(lim sup bn)
2) If {an} and{bn} are non-negative sequences and {bn} converges, prove that lim sup anbn = (lim sup an)(lim bn).
I am not sure how to show these two similar proofs. Please show details so I can understand the process.
Thank you.
Let be $n\geq0$. For all $k\geq n$ , one has$$a_{k}\leq\sup_{k\geq n}a_{k}$$ and also$$b_{k}\leq\sup_{k\geq n}b_{k}.$$ Since everything is non-negative, one gets$$a_{k}b_{k}\leq\left(\sup_{k\geq n}a_{k}\right)\left(\sup_{k\geq n}b_{k}\right)$$ whence$$\sup_{k\geq n}a_{k}b_{k}\leq\left(\sup_{k\geq n}a_{k}\right)\left(\sup_{k\geq n}b_{k}\right).$$ To conclude, one takes the limit $n\rightarrow+\infty$.