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 $L$