I have a simple statement about sequences I have proven. I am asking for help to see if this statement and proof are correct. If statement is correct, is there simpler way proving it? Huge thanks!!!! Here it is:
Statement: ${\{a_n}\}$ is a Cauchy sequence in R. Let $S_K = \{x=d(a_m,a_n)/m,n>K,K\in N\}$. Here $d(\cdot,\cdot)$ is a distance metric. Also $S_{K+1}\subset S_K $.
Prove: if ${\{a_n}\}$ diverges, then $\exists\epsilon>0\ni\forall K[\forall x\in S_K(x\geq\epsilon)]$. In words, if the sequence diverges then all the distances are larger then certain $\epsilon$.
Proof:
(1) By Cauchy criterion: if $\{a_n\}$ diverges then $\exists\epsilon>0\ni \forall K \exists m,n\geq K \implies d(a_m,a_n)\geq\epsilon $. Rewriting this statement in terms of $S_K$: if $\{a_n\}$ diverges then $\exists\epsilon>0\ni \forall K \exists x\in S_K \implies x\geq\epsilon $. It follows then that $\exists\epsilon>0\ni \forall K (\sup S_K \geq\epsilon)$
(2)To prove the main statement we have to show that:
$\forall x \in S_1 \exists V\ni x\geq \sup S_V$. (aaa)
Take an arbitrary $x\in S_1$, first notice that $\exists U \ni x\in S_U\subset S_1$. There are two ways in which (aaa) is not true. (1) x is smallest element $\implies \bigcap{_{n\in N}}S_n = {\{x\}}$ which is not true since $n\rightarrow\infty $ and intersection should be empty. (2) $\forall S_K(S_K\subset S_U$ and $x\notin S_K)$ has no supremum which contradicts Least Upper Bound property of real numbers. Hence (aaa) is correct and main statement is true.