Ytukyg

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 $$a_ngeq (n+2) lambda-g(n)$$

Any tips on how to proceed? Thanks in advance.