Asymptotic notation propertiesLet and be asymptotically positive functions. Prove or disprove each of the following conjectures.

- implies .
- .
- implies , where and for all sufficiently large .
- implies .
- .
- implies .
- .
- .

- Let and . Hence, but .
- Take the same example as above. .
- implies for all such that the constants .
Hence, .
Therefore, .
- Let and . Hence, but .
- If for sufficiently large values of , , i.e. . However, if , this conjecture does not hold.
- implies for all such that the constants . This inequality can be rearranged as , i.e. .
- Let . .
- From definition, . Hence, . Therefore, .