How to prove that $a/2 < a$ for all $a > 0$. I am working on another problem and used proof by contradiction to arrive at $a < a/2$. However, I do not know if I can say that this obviously cannot happen for $a \gt 0$ or if I need to prove this statement. I am new to real analysis and still really unsure of what needs to be proven and what can be taken as a given. Thanks.
 A: It honestly depends on the rigor that is expected of you - this isn't something you should really address to MSE, but rather to your professor. See how they feel about the level of rigor, and about the "obviousness" that one half of a positive value is less than the value itself.
Generally, at least in my experience, it was taken as obvious. You could probably prove it relatively simply if you wanted:
$$a > 0 \implies a + a > a \implies 2a > a \implies a > a/2$$
So it's not like it would raise a huge fuss if you tried to prove it, considering it would be relatively little space on the paper, and little effort on your part. But at the same time, confusions like these are absolutely understandable and will likely be a repeating problem in the course, but the best you're really able to do is talk to your professor and see what they consider "kosher."
A: I don't think it is really worth to prove that, as it is pretty immediate. In practice, anyone able to follow a proof will accept with this fact.

A valid proof is:
For $a>0$,
$$\frac12<1\iff \frac a2<a$$ using the rule of factors applied to an inequality.
