# Decide if a language has a word of a given size

Suppose that $$L$$ is some language over the alphabet $$\Sigma$$. I was asked to show that the following languages is decidable:

$$L’ = \{w \in \Sigma^* | \text{ there exists a word } w’\in L \text{ such that } |w’| \leq |w| \}$$

I.e., $$w \in L’$$ if $$L$$ has some word with length smaller than $$|w|$$.

The way I was thinking to show that is observing that $$L \cap\Sigma^{|w|}$$ is finite, and $$(L \cap \Sigma) \cup (L \cap \Sigma^2) \cup \ldots\cup (L\cap \Sigma^{|w|})$$ is finite too, hence decidable. But the main thing I am struggling with is how can any algorithm for $$L’$$ know if some $$u \in L$$? this is undecidable, so it’s unclear to me how any algorithm for $$L’$$ can verify that indeed some word is in $$L$$