show that in every infinite computably enumerable set, there exists an infinite decidable set
Here is one possible approach. Since $L$ is c.e., there is some enumerator that outputs a list of the word in $L$: $w_1,w_2,\ldots$.

Let $D$ consist of all words $w_i$ which are longer than all words appearing before them. I claim that $D$ is infinite. If not, let $w_m$ be the last word in $D$. Then all other words have length at most $|w_m|$. However, this contradicts the infinitude of $L$.

Furthermore, $D$ is decidable. Given a word $w$, run the enumerator for $L$, and halt whenever one of the following events happen:


If $w_i = w$, output Yes.
If $|w_i| &gt; |w|$, output No.


Since $L$ is infinite, this algorithm will eventually stop.