Question

What does bounded above or below mean in precalculus?

Answer 1

Hint: Here we will discuss a very important topic of pre calculus which are bounded above and bounded below. This bounded above and bounded below is used in series and sequence of the function. And sometimes we use these two to find the limit of the function. Actually these two are the starting topics of real analysis or pre-calculus when we study the sequence and series of the function.

Complete step by step solution:
Now to understand these terms we will understand it by taking example because without example it is hard to understand the definition of the bounded above and bounded below. But before taking an example, let me tell you the definition of bounded above and below.
A set is bounded above by the number A if the number A is higher than or equal to all elements of the set.
A set is said to be bounded below by the number B if the number B is lower than or equal to all elements of the set.
Here set is a collection of distinct elements.
Now to understand both these definition let`s take an example,
Let A be a set such that$A=\{{\displaystyle \frac{1}{n}}:n\in \mathbb{N}\}$
Since the entries are coming from natural number so we can write above set as also $\Rightarrow A=\{1,{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{3}}.........\}$
We can see easily that $1>{\displaystyle \frac{1}{2}}>{\displaystyle \frac{1}{3}}>....$ , it means $1$ is the greatest number in the above set. So by definition of the bounded above it is clear that $1$ is bounded above.
Similarly, we can see that $1>{\displaystyle \frac{1}{2}}>{\displaystyle \frac{1}{3}}>....$ it is tending to zero, it means the last of this sequence is greater or equal to zero, but there is no number which is less than zero so by the definition of the bounded below we can easily concluded that $0$ is bounded below of the set.
Hence, for $x\in A$ we can write $0\le x\le 1$.

Note: It is not hard to find the bounded above and bounded below of the given set. But sometimes the sets are too complicated and it's hard to find the bounded above and bounded below, then we have to simplify the set then we get our better results. Sometimes it also happens that some sets are only either bounded above or bounded below. Like $[0,\mathrm{\infty })$ this set is only bounded below but not bounded above. So we do not say the set is bounded, we also say the set is bounded below not bounded above.