I'm not clear on the definition of sequence such that you could have one without having a description of how to produce it? Can you have a sequence of random numbers?
You're kind of "begging the question" here, where you're assuming that non-computable numbers exist and then using that to show that some numbers are non-computable. You can definitely show that these things exist, but that relies on "believing" the set of axioms that you used to prove it.