I know that $L=\{ \langle M \rangle \mid |L(M)| < \infty \}$ is not decidable (by Rice's theorem or using reduction, I followed it from [$L$ not being decidable][1] ).  But is $L$ recognizable?

What I tried is, let $L$ have a machine that recognizes it, let it be called $H$. Then given an input $\langle M\rangle$ I would start enumerating all strings in $L$ by using $H$. As $L$ has infinite many strings at some point the string being enumerated will be equal or larger than $\langle M\rangle$ (in lexicological order), thus using $H$ I am able to decide $L$ which I know is not possible.

Is my method correct ? In either case is there a better method for example is there a  general way for proving that a language is not recognizable like for undecidability  we try to reduce an undecidable problem to the current problem ? 


  [1]: https://cs.stackexchange.com/questions/24439/can-you-recognize-or-decide-if-a-turing-machine-has-an-infinite-sized-language