Here is a more readable version of the question:

Prove that for all functions $ g: \mathbb{N}\to\mathbb{R}^{\geq 0}$ , and all numbers $ a \in \mathbb{R}^{\geq 0}$ , if $ g \in \Omega(1)$ then $ a + g \in \Theta(g)$

What I’ve done so far:

In order for $ a + g\in\Theta(g)$ , $ a+g \in \mathcal{O}(g) \wedge a + g \in \Omega(g)$ .

Expanding our assumption, we get:

$ \exists c_1, n_1 \in \mathbb{R}^{+}, \forall n \in \mathbb{N}, n \geq n_1 \implies g(n) \geq c_1$

**Proving $ a + g \in \Omega(g)$ **:

Expanding the definition, we get:

$ \exists c_2,n_2 \in \mathbb{R}^{+}, \forall n \in \mathbb{N}, n \geq n_2 \implies a + g(n) \geq c_2*g(n)$

Let $ c_1 = 1$ and $ n_2 = n_1$ . Let $ n \in \mathbb{N}$ . Assume $ n \geq n_2$ . Prove $ a + g(n) \geq c_2 * g(n)$ .

$ g(n) = g(n)\ \implies g(n) \geq g(n)\ \Leftrightarrow c_1 * g(n) \geq c_1 * g(n)\ \Leftrightarrow g(n) \geq g(n) \text{ (since $ c_1 = 1$ )}\\Leftrightarrow a+g(n) \geq g(n) \text{ (making left side bigger since $ a \in \mathbb{R}^{\geq 0}$ )}$

**Proving $ a+g \in \mathcal{O}(g)$ **:

Expanding the definition, we get:

$ \exists c_3,n_3 \in \mathbb{R}^+, \forall n \in \mathbb{N}, n \geq n_3 \implies a + g(n) \leq c_3 * g(n)$

I’m struggling here since I’m not really sure what value of $ c_3$ I should be using or how to derive one. (I’ve tried using my assumption of $ g(n) \geq c_1$ but I don’t really know where to go from there). Any help is greatly appreciated and I apologize for any formatting errors in advance. Thank you.