elementary number theory - Validity of Inductive Proof - Proof Confirmation

I want to prove this statement using weak induction:

Every integer $n>11$ is a sum of two composite integers.

When I prove it I get stuck at something basic I believe but unclear for me:

I prove it separately for the odd $2n+1$ and the even $2n$ numbers:

$n$ must be: $n\ge6 (2*6>11)$

I'm stuck at the same thing for odd and even numbers so I'll ask it regarding the even ones.

I check the validity of the claim for the basic cases when $n=6$.
$2*6=12=6+6$. Indeed is a sum of two composites.

Then I assume it's true for $2n$ and check for $2(n+1)$:

Then i get stuck because $2(n+1)$ and $2n$ are the of same form.

So can I assert the claim is true for $2(n+1)$ because it's the same form as $2n$ and based on my induction assumption earlier its true for $2n$?

Answer

$2n = j + k$ ($m$ and $n$ composite)

$2(n + 1) = j + k + 2$.

If $j$ (or $k$) is even (well, if one is even they both are but I'll pretend we don't know that) then $j + 2$ (or $k + 2$) is even and thus composite. $2(n + 1) = (j + 2) + k$ (or $j + (k + 2)$); sum of composites.

If $j$ and $k$ are both odd then $j + 1$ and $k + 1$ are both even so $2(n + 1) = (j + 1) + (k + 1)$; sum of composites.