while this method is perfectly valid way to show the lower bound, I would like some help in filling out the following, different argument: Suppose I am given a subset of N, or an element in P(N). My map is as follows: I order this set, and make, something akin to a cycle (if the set I pick happens to be finite, this is pretty well defined i guess, I just put 2 brackets at the start and the end of the finite set and call it a cycle). If the set is infinite, I follow the order, as in, if the set is (1,3,5,7, ...),

I have this theorem: "If a sequence is unbounded above, then there exists a monotonic subsequence that tends to $+\infty$". In the proof, the author use the unboundedness to say that there exists $n_1, n_2 \in \mathbb{N}$ such that $a_{n_1}>0$ and $a_{n_2}>\text{max}(a_1,\dots,a_{n_1},1\}$. Then, the author says that it must be $n_2>n_1$. Finally, we can iterate the construction and obtain $a_{n_{k+1}} > \text{max}(a_1,\dots,a_{n_k},k\}$.

I have two questions, first one: is the $a_{n_1}>0$ condition just because it is how the author wants to construct the subsequence with the $k-1$ as last…

Maybe theyre not obvious? have u considered that? For example, things like $x^y$ where x and y are real numbers are not really natural objects like we used to think of them in school etc. in high school we just without thought used things like $e^x$. after my 1st analysis course I understood its not a "natural" thing (being intentionally vague). its the supremum of the set of all x^t where t runs over $\mathbb{Q}$ such that t <= y. later on itl be a taylor expansion, and so on.

Also, "obvious" statements like, say, given $y>0$ and $n \in \mathbb{N}$, there exists $x \in \mathbb{R}^+$ such …