## \$G = ({, , …})\$ infinite turing recognizable language, prove \$(L(M1) ∪ L(M2) ∪L(M_3).. )\$ is itself turing recognizable?

Given that $$G = ({, , …})$$ is an infinite turing recognizable language, whose all inputs are descriptions of turing machines,

how can one prove that the union of all the languages recgonized by the turing machines in $$G$$: $$(L(M1) ∪ L(M2) ∪L(M_3).. )$$ is itself turing recognizable?

I tried to prove it by building an enumerator for the language, using the fact that since $$G$$ is turing recognizable it can be produced by one.

``E' =  1. Initialize steps counter i=0 2. Run E, the enumerator of T, number of steps equal to i. 3. For every turning machine description M produced by E:    3.1 for every input word corresponding to 0 up to i (according to lexicographic order do:    3.2 Run M on the current input.    3.3 print if M accepts. 4. increase i by 1, and go back to step 2. ``

Is this a correct approach? Is $$E’$$ valid and producing the language as needed? If not, what am I doing wrong?

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