I'm reading Computation Complexity: A Modern Approach and one of the exercises is:
Prove that [the addition function is] computable by writing down a full description (including the states, alphabet, and transition function) of the corresponding Turing [machine].
I successfully wrote this description but I don't think it is a proof the addition function is computable.
According to this question, the definition of a computable function is:
If $f:\Sigma^* \to \Sigma^*$ is function, and $\exists$ a Turing machine which on the input $w\in\Sigma^*$ writes $f(w)$, $\forall w\in\Sigma^*$, then we call $f$ as computable function.
With the Turing machine I wrote, I can prove the machine is able to compute $f(w)$ for some inputs but how can I prove it can do the same for every inputs $w\in\Sigma^*$? In other words, how can I prove $f$ is computable?