1. **Stating the problem:** We are given the interval $[1, 2] \subseteq \mathbb{R}$. We need to find the upper and lower bounds of this interval. Next, determine how many upper and lower bounds exist, find the least upper bound (supremum) and greatest lower bound (infimum) from these bounds. Finally, guess the number of integers, rational numbers, and real numbers inside this interval. 2. **Find upper and lower bounds:** - The interval $[1, 2]$ includes all real numbers $x$ such that $1 \leq x \leq 2$. - Therefore, any number $\geq 2$ is an **upper bound** for this interval. - Any number $\leq 1$ is a **lower bound** for this interval. 3. **How many upper and lower bounds are there?** - Since every number greater than or equal to $2$ is an upper bound, there are infinitely many upper bounds. - Similarly, every number less than or equal to $1$ is a lower bound, so infinitely many lower bounds exist. 4. **Find the least upper bound and greatest lower bound:** - The **least upper bound** (supremum) is the smallest number that is an upper bound. This is $2$. - The **greatest lower bound** (infimum) is the largest number that is a lower bound. This is $1$. 5. **Guess the number of integers in $[1, 2]$:** - The integers in the interval $[1, 2]$ are $1$ and $2$. - So, there are exactly $2$ integers. 6. **Guess the number of rational numbers in $[1, 2]$:** - Between any two real numbers, there are infinitely many rational numbers. - Hence, infinitely many rational numbers lie in $[1, 2]$. 7. **Guess the number of real numbers in $[1, 2]$:** - The interval $[1, 2]$ contains uncountably many real numbers. - Thus, there are infinitely many (uncountably infinite) real numbers in the interval. **Final answers:** - Upper bounds: infinitely many, least upper bound = $2$ - Lower bounds: infinitely many, greatest lower bound = $1$ - Integers in $[1, 2]$: $2$ - Rational numbers in $[1, 2]$: infinitely many - Real numbers in $[1, 2]$: uncountably infinitely many