Asymptotic notation properties

Let and be asymptotically positive functions. Prove or disprove each of the following conjectures.

  1. implies .
  2. .
  3. implies , where and for all sufficiently large .
  4. implies .
  5. .
  6. implies .
  7. .
  8. .
  1. Let and . Hence, but .
  2. Take the same example as above. .
  3. implies for all such that the constants . Hence, . Therefore, .
  4. Let and . Hence, but .
  5. If for sufficiently large values of , , i.e. . However, if , this conjecture does not hold.
  6. implies for all such that the constants . This inequality can be rearranged as , i.e. .
  7. Let . .
  8. From definition, . Hence, . Therefore, .
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