$a_{n+m+2} \leq a_m+a_n+g(n)$ with $g(n) = o(n)$. Show that $a_n \geq (n+2)\lambda-g(n)$ where $\lambda = \lim \frac{a_n}{n}$

Question

I was working on the proof of the exponential decay on supercritical percolation as shown in Grimmett's Percolation (1999, 2 ed. pg 206 - 210) and he uses as a lemma a form of the subadditive theorem (fekete) in which he states (he says it is easy to find) that

if you have a sequence $(a_n)_n$ such that $a_{n+m+2} \leq a_m+a_n+g(n)$ and $\lim_{n \to \infty} \frac{g(n)}{n}=0$ then

• the limit $\lim_{n \to \infty} \frac{a_n}{n} = \lambda$ exists and it's equal to the $\inf \left\{\frac{a_n}{n}\right\}$.

• $a_n \geq (n+2)\lambda-g(n)\;\forall n$.

Now, although I coundn't prove the existence of the limit using only what was stated on the theorem, I manage to prove it adding some restrictions which are valid on Grimmet's theorem (I asked it in here: Subadditive lemma for $a_{m+n+2}\leq a_n+a_m+g(m)$ and I found that, in my application, the last condition was satisfied).

Now assuming the limit exists, I couldn't manage to show that $$a_n\geq (n+2) \lambda-g(n)$$

Any tips on how to proceed? Thanks in advance.