I have been trying to figure out(prove) Goldbach Conjecture(Strong) which states: Every even integer greater than 2 can be expressed as the sum of two primes. My question I guess is general, is it wrong for me to prove something using "simple statements". Here are my statements:
Every prime number greater than 2 is odd.
Every even integer is the sum of two odd numbers.
Therefore, the conjecture is true.
In math, is it not allowed to make general statements OR must I prove this by other means like for example, proof by contradiction, direct proof etc.?
In general, proving something using "simple statements" is not only acceptable, but encouraged - the best proofs are the ones that use your general format. The thing is, your series of simple statements do not form a proof. In a proof, each statement must be a consequence of the one before; your first two sentences do not entail the third. The easiest way to see this is that, for example, $42$ can be written as $21 + 21$, the sum of two odd numbers. But $21$ is not prime, so $21 + 21$ isn't a way of writing $42$ as a sum of two primes. Now, we can also write $42$ as $19 + 23$, but the point is that the existence of a way to write it as a sum of two odd numbers doesn't tell us how to write it as a sum of two primes.