Well done sir. I also appreciated your inclusion of proper reading and writing of the math. The confidence gained and utility gained of writing math should not be underestimated. Few things will help you feel like a real math insider more than writing your math. (I like and subscribe to MAT137)
You are not trying to prove the sequence is convergent. You are assuming the sequence is convergent. So we know that it is true that "for every epsilons... (something)". Thus, if you pick any value of epsilon you want, the "(something)" part will be true.
