Is $$f(n) \in \Theta(g(n)) \Leftrightarrow f(n) = g(n) \cdot (1+o(1))$$true?For clarity, here are the definitions I use:$$f(n) \in o(g(n)) \Leftrightarrow \forall \epsilon > 0 \exists n_o\forall n> n_0: |f(n)|\leq \epsilon \cdot |g(n)|$$$$f(n) \in O(g(n)) \Leftrightarrow \exists c > 0 \exists n_o\forall n> n_0: |f(n)|\leq c \cdot |g(n)|$$$$f(n) \in \Omega(g(n)) \Leftrightarrow \exists c > 0 \exists n_o\forall n> n_0: c \cdot |g(n)| \leq |f(n)|$$$$f(n) \in \Theta(g(n)) \Leftrightarrow f(n) \in \Omega(g(n)) \land f(n)\in O(g(n))$$What I have so far:The $\Leftarrow$ direction is quite easy:$$f(n) \in o(g(n)) \Rightarrow f(n) \in O(g(n)) \land g(n)+o(1)\cdot g(n) \in \Omega(g(n))$$However, I am unsure about the other direction. I suspect that it's not, but can't proof it.PS: This is not a homework assignment, but homework related: I am working on a seminar paper and want to understand this.
Relationshop between Theta and little-o (Big-O notation)
asymptotics