Prove Σ* is decidable
Theorem: The set $\Sigma^{*}$ of all words is decidable.

Proof. According to the definition of decidability, we must provide a computable function $d$ which takes a word $w$ and outputs $1$ if $w \in \Sigma^{*}$, and outputs $0$ if $w \not\in \Sigma^{*}$. Such a function is very easily constructed, it is
$$d(w) = 1,$$
That is, because every word is in $\Sigma^{*}$, the decision function always says "yes". QED.