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Homework Statement
Find sup A if A = {0.2, 0.22, 0.222, 0.2222, ...}
I'll write elements of a set with low case letters and indexes, e.g an
The Attempt at a Solution
Begin by definition of supremum:
[itex]\sup A = a[/itex] if [itex]\forall x \in A, x \leq a[/itex] and [itex]\forall b \in \mathbb R ((\forall x \in A , x \leq b) \rightarrow a \leq b[/itex] essentially, a supremum is the lowest value of upper bounds of a set.
Also if [itex]\exists \max A \rightarrow \sup A = \max A[/itex]
So I'll try to find a max A and prove sup A exists:
Let us have [itex]\exists \max A = M[/itex] such that [itex]\forall x \in A, x \leq M[/itex]
Suppose that [itex]a_n = M[/itex]. [itex]a_n = \sum_{k=0}^n\frac{1}{5 \cdot 10^k}, n \in \mathbb N[/itex]. If [itex]a_n \geq a_{n+1}[/itex] then [itex]\max A = \sup A = a_n[/itex]
[itex]a_{n+1}= \sum_{k=0}^{n}(\frac{1}{5 \cdot 10^n})+ \frac{1}{5 \cdot 10^{n+1}} \rightarrow a_{n+1} > a_n[/itex]. Therefore max A does not exist.
However, I cannot conclusively prove that sup A does not exist at all for this set A. Intuitively I can see that the elements' difference is becoming smaller and smaller and smaller, hence they should eventually be limited to some specific value. Altho it says if max A exists, it's also the sup A, but it does not say if max A doesn't exist, then there is no sup A.
What am I missing?
Thanks in advance.
EDIT:
Quick and dirty - If I assumed sup A = 0.23, for example, would it be sufficient evidence that sup A does not exist if I show there is a value 0.223 which is still greater than all the elements in the set A, but lesser than the supposed 0.23. Then it follows that I can suppose sup A = 0.222223, which still satisfies the upper bound criteria, however is not the least of the upper bounds. How can I show that there is no sup A in this case?
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