So, I have just started self-studying Analysis, and I'm constantly running into the same problem. It's better that I illustrate the issue using an example. $$(\forall n \in \mathbb{N}) (\forall a \in A) \space s + \frac{1}{n} \geq a \implies s +\frac{1}{n} \gt a \space \vee \space s+\frac{1}{n}=a $$ This is obviously the definition of an upper bound.
What I have noticed is that using an implication like this tends to make the problem a lot easier--given that there is a disjunction in the consequence of the implication. This makes me think that I have to prove only one of the two consequent arguments, but I feel very wary about doing so and avoid resorting to it. Is my apprehension warranted, or am I overthinking needlessly?
Also, for this particular implication, I wanted to ask how the quantifiers might carry over to the resulting arguments. Would they remain intact or might they change?
Since ≤ in words means less or equal I don't see the difference of just using it.Usually the = is easy here if s=a-1/n and the < is harder to prove. like in $\sqrt{ab}≤\frac{a+b}{2}$ the = is for a=b the < needs some effort,you should not prove only one. what you mean for the quantifiers I don't know, quantifiers ar for variabkes not arguments.