epsilon-delta proofs 3

In [Def-1], we defined that a sequence of real numbers $a_n$ approached $a$. The definition that will bring mathematical complications to us has the advantages which make us possible to discuss precisely. Let a sequence $b_n$ be as follows.

$b_0=2.3$
$b_1=2.38$
$b_2=2.388$
$b_3=2.3888$
・・・・・・

Obviously in this sequence, as $b_{n+1}$ is always greater than $b_n$, the greater $n$ becomes, the greater $b_n$ becomes. Hence, it is not wrong that $b_n$ comes closer to $a=2.4$. But no one will be convinced that $b_n$ approaches $a=2.4$. The differences of $|b_n-a|$ are

$|2.3-2.4|=0.1$
$|2.38-2.4|=0.02$
$|2.388-2.4|=0.012$
$|2.3888-2.4|=0.0112$
・・・・・・

Look at above and guess more. No matter how small the differences are, it is definitely no doubt for us to understand that those do not become less than $0.01$, and $b_n$ does not come closer to $2.4$ over $2.39$. We never want to include a kind of sequences $b_n$ in sequences approaching $a=2.4$.

On the preceding calculation, we put $b=2.3888\cdots $, and multiply by $10$. Then, as $10b=23.888\cdots $, after subtracting both sides of first equation from this, we get a rational number $b=43/18$. It means that if $n\rightarrow \infty $, $b_n$ approaches $b=43/18$. $b$ is not $432/180=2.4$.

If we accept a definition of converging sequences by saying more and more or lower and lower, we also cannot eliminate from sequences approaching $a=2.4$ as follows.

$c_0=0.3$
$c_1=0.33$
$c_2=0.333$
$c_3=0.3333$
・・・・・・

We enough know that $c_n$ approaches $c=1/3$. Even these examples, it is clear that we have to eliminate such a kind of sequences by adopting a precise definition of converging sequences. This is [Def-1].

If  we conform to [Def-1], for any $\epsilon >0$ less than $0.009$, as we are not able to find out a $N>0$ such that if $n\geq N$, $|a_n-a|< \epsilon$. As it goes against [Def-1], we can eliminate $b_n$ from the class of $a_n$. We can do $c_n$, too.

This is a reason why epsilon-delta definitions are nice, and we have to adopt [Def-1], although it's much complicated.