real number system 9 (bounded monotone sequences)

In the method of nested intervals, we supposed the following proposition.

If a real number sequence increasing or decreasing monotonically is bounded,  it will converge.

We shall prove it simply. A monotonical sequence is either

$(increasing)\quad a_1\leq a_2\leq\cdots\leq a_n\leq\cdots$ , or
$(decreasing)\quad a_1\geq a_2\geq\cdots\geq a_n \geq\cdots$ .

We will adderess the case of "a increasing sequence", as both cases are same.

If a increasing sequence is bounded above, it has $\sup a_n=A$ (by the real number system property). Therefore for any $\epsilon>0$,
\[  A-\epsilon < a_m\leq a_{m+1}\leq \cdots \leq A  \]
Namely for all $n$ such that $n>m$ , $|a_n-A|<\epsilon $  because $A-\epsilon<a_n\leq A$ .
As it means $\lim_{n\rightarrow\infty}a_n=A$ , we get the desired result.