(8 points) Using the definitions for asymptotic notations:\ i) Prove that

5lgn=o(nlgn)

\ ii) Prove that

2lgn=O(\\\\sqrt(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.\ Also, recall from the lectures that

lg(n)

is same as

log_(2)(n)

.

5. (8 points) Using the definitions for asymptotic notations: i) Prove that 5lgn=o(nlgn) ii) Prove that 2lgn=O(n) Note that for your proofs, you will have to show the existence of the two constants c and n0 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 n0 that works, like I did during the lecture time at least for one example. 2 Also, recall from the lectures that lg(n) is same as log2(n)