Suppose M accepts a language L Sigma.
  Let w_1, w_2, ... be all strings over Sigma, in canonical order.

  The following program enumerates L:

  for all t from 0 to infinity
   for all i from 1 to t
     If M accepts w_i$ within t steps
       write w_i

  By the Church-Turing thesis, some Turing machine accepts L.

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  Suppose M enumerates a language L over Sigma.
  Let w_1, w_2, ... be the enumeration of L given by M.

  The following program accepts L:

  Read a string w in Sigma*
  for all i from 1 to infinity
    if w = w_i
      write "Yes"
  
  By the Church-Turing thesis, some Turing machine enumerates L.