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Do the following implications hold, enter image description here

I can see that the reverse implication will be true and these implications seem to be true as well, but I can't think of a way to prove or disprove it. Can anyone think of a counter example?

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    Please put all relevant info in the question directly. – 2017-01-26

1 Answers 1

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$f(n) = \Theta(g(n))$ implies there exist constants $a$ and $b$ such that $$a \le \frac{f(n)}{g(n)} \le b$$ for all large $n$.

Then, $$\left|\frac{f(n)-cg(n)}{g(n)}\right| = \left|\frac{f(n)}{g(n)} - c\right| \le \max\{|a-c|, |b-c|\} =: C$$ for all large $n$.

Counterexample for a):

Let $f(n):= (-1)^n$ and let $g(n)=1$.