Using the definitions for asymptotic notations:

i$)$ Prove that $5$ lo${}_{}$g2 n $=o($n lg n$)$

ii$)$ Prove that $2$ lo${}_{}$g2 n $=O($n$)$

Note that for your proofs, you will have to show the existence of the two

constants c and n$0$ as per the respective asymptotic $($O or o$)$ notation$$s

definition. If needed, it is okay if you end up using a spreadsheet

calculator to compare the two functions, in order to find a combination

of c and n$0$ that works, like I did during the lecture time at least for

one example.