1. **Problem Statement:** Prove that the set of real numbers is uncountable. 2. **Key Concept:** A set is countable if its elements can be put into a one-to-one correspondence with the natural numbers $\mathbb{N}$. Otherwise, it is uncountable. 3. **Approach:** We use Cantor's diagonal argument to prove that the real numbers between 0 and 1 are uncountable, which implies the entire set of real numbers is uncountable. 4. **Cantor's Diagonal Argument:** - Assume, for contradiction, that the real numbers in the interval $(0,1)$ are countable. - Then, we can list all such numbers as $r_1, r_2, r_3, \ldots$ where each $r_i$ is represented by its decimal expansion. 5. **Construct a new number:** - Create a number $r$ whose $i$-th decimal digit differs from the $i$-th decimal digit of $r_i$ (for example, if the $i$-th digit of $r_i$ is 5, choose 6 for $r$'s $i$-th digit; otherwise, choose 5). 6. **Contradiction:** - By construction, $r$ differs from every $r_i$ in at least one decimal place. - Therefore, $r$ is not in the list, contradicting the assumption that all real numbers in $(0,1)$ were listed. 7. **Conclusion:** - Hence, the set of real numbers in $(0,1)$ is uncountable. - Since $(0,1) \subset \mathbb{R}$, the entire set of real numbers $\mathbb{R}$ is uncountable. This completes the proof.