BMCT Convergence 
Given $x_1 = 3$ and $x_{n+1}=\frac{1}{4-x_n}$, show that $\{x_n\}_{n\geq 1}$ is  decreasing and bounded below by $0$ and above by $4$.

In this case, can we say that it diverges by BMCT ?? because it should be bounded below $or$ bounded above, not both right?
 A: This is extremely important to remember. In math, the word "or" is inclusive. 
This means that if I say $A\;\text{or}\; B$, I mean to say that there are three possibilities: $A$, $B$, or both. 
In ordinary language, the word "or" is ambiguous because people use the word to mean different things at different times. In mathematics, logic, and computer science, however, people try to make things as precise as possible. For this reason, people in computer science use the word "or" to explicitly mean $A$, $B$, or both, and the word "xor" to mean $A$, $B$, but not both.
So when the word "or" is used, you only need to satisfy at least one of the conditions. If one of the conditions is met, you don't care about the rest.

About the bounded monotone convergence theorem, you can think of the theorem as saying: if 


*

*$(x_{n})$ is monotonically increasing and bounded above, OR

*$(x_{n})$ is monotonically decreasing and bounded below,


then $(x_{n})$ is convergent. If we understand the percise meaning of "or", then we understand that convergence may come from the first bullet point, the second bullet point, or both. At least one of the conditions may be satisfied to get convergence.
