Sorry if its pretty basic, I just wanna know if I did it right! I would love any feedback on it! Was there excessive details, or too little?
Proof:
let $A_n$ be any nondecreasing, bounded sequence. Since it is bounded, it has a supremum $a$. let $\epsilon$ be any positive number. By the definition of supremum, there exists $A_v$ > $\alpha-\epsilon$. Since the sequence is nondecreasing, for all $x$ in $\mathbb N$ ,$x\geq v$ , then $a-\epsilon<A_x \leq a<a +\epsilon$. Therefore, $\forall \epsilon >0, \exists v \in \mathbb N$ such that $\forall x \in \mathbb N, x \geq v$ , |$A_x-a$|<$\epsilon$. Thus, the sequence converges to its supremum $a$.
Best Answer
In the second line, you have $\alpha$ instead of $a$.
Moreover, in the fourth sentence, I would write: "there exists $v \in \mathbb N$ such that $A_v > a - \epsilon$".
Otherwise, the proof seems to be fine. The amount of detail is ok.