Infinite languages and undecidable languages

I'm having a problem with proving the following statement:

For every infinite language $L$, does there exists an infinite language $L' \subseteq L$ such that $L'$ is not decidable?

The set $\{L' : L' \subseteq L\}$ is an uncountable set of languages, and since there is only countably many decidable languages, it has to contain an undecidable language. (You can get a concrete $L'$ by a diagonal argument.)

StackExchange Computer Science Q#12309, answer score: 7