Question

What is boundedness?

Answer 1

Hint: Boundedness is also termed as bounded above or bounded below. Consider S to be a set of real numbers. S is called bounded above if there is a number M so that any $x\in S$ is less than, or equal to $M:x\le M$ . The number M is called an upper bound for the set S. S is called bounded below if there is a number m so that any $x\in S$ is greater than or equal to $m:x\ge m$ . The number m is called a lower bound for the set S.

Complete step by step solution:
Let us learn about boundedness. Boundedness is also termed as bounded above or bounded below.
We can first see the bounded above. Consider S to be a set of real numbers. S is called bounded above if there is a number M so that any $x\in S$ is less than, or equal to $M:x\le M$ . The number M is called an upper bound for the set S. In other words, we can say that a set S is bounded above by a number M, if M is greater than or equal to all the elements of S. If M is an upper bound for S then any bigger number ( $M\ge x$ ) is also an upper bound.
Now, let us see the bound below. S is called bounded below if there is a number m so that any $x\in S$ is greater than or equal to $m:x\ge m$ . The number m is called a lower bound for the set S. In other words, we can say that a set S is bounded below by a number m, if m is less than or equal to all the elements of S. if m is a lower bound for S then any smaller number ( $m\le x$ ) is also a lower bound.
Now, let us consider an example of set $A=\left\{ \dfrac{1}{n},n\in N \right\}$ . We can write this set as $A=\left\{ 1,\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},... \right\}$ . We can see that the maximum value of the set is 1. So if we consider any real number, M, this M will be greater than or equal to the element of S, that is, 1. Therefore, we can say that $M=1$ is an upper bound of the set A. Also any real number $M\ge 1$ is also an upper bound of the set A. Similarly, we can say that the set is tending to 0. Therefore, the lower bound will be 0.
Therefore, we can write $0\le x\le 1,x\in A$ .

Note: Students must not get confused with upper bound and lower bound. There are some sets which do not have any of these. Let us consider a set $A=\left\{ 1,2,3,.. \right\}$ . This set is bounded below by 1, but not bounded above.