Prove statement- Little-O notation

[evinda]

Hello! (Smile)

I want to prove or disprove the statement $f(n)+o(f(n))=\Theta(f(n))$.
Do I have to set $g(n)=o(f(n))$ and use the definition, that is that $\forall c>0, \exists n_0(c) \in \mathbb{N}$ such that $\forall n \geq n_0: 0 \leq g(n)<cf(n)$ ? (Thinking)

Is the following right?

We set $g(n)=o(f(n))$.That means that $\forall c>0, \exists n_0 \in \mathbb{N}$ such that $\forall n \geq n_0: 0 \leq g(n)< cf(n)$.
So: $f(n) \leq f(n)+g(n) < cf(n)+f(n)=(c+1)f(n), \forall n \geq n_0$.
Right? So do we say that $f(n) \leq f(n)+g(n) < cf(n)+f(n)=(c+1)f(n) \Rightarrow f(n) \leq f(n)+g(n) \leq (c+1)f(n), \forall n \geq n_0$

So picking $c_1=1, c_2=c+1$ we see that $f(n)+g(n)=\Theta(f(n)) \Leftrightarrow f(n)+o(g(n))=\Theta(f(n))$.

 

[alyafey22]

I think the statement is wrong by counter example.

 

[evinda]

I think the statement is wrong by counter example.

So is my proof wrong? Which counterexample did you find?

 

[alyafey22]

So is my proof wrong? Which counterexample did you find?

I'm really sorry. I think I am wrong. Your proof is correct.