I am starting analysis and one of the problems I am encountering is since I do not have a teacher or guide I am not sure if what I write is sufficent or overly verbose etc. I would appreciate if anyone would critique this solution to a problem and maybe give me suggestsions such as "You cannot assume this" or "You needn't state that", things of that nature. Thank you very much in advance.
Question: "Fix . If , define to be the set of all numbers , where is rational and . Prove that:
Solution: . Furthermore if (Can I assume this, or would I have to prove it?). So we can state that . So is bounded above by . And if would not be an upper bound of . Therefore
Is that ok? If it's horrific I apologize, it's my first time doing these kind of questions.