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.
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