2. Let $a_{1}, a_{2}, \ldots, a_{n} \in \mathbb{R}$. Answer the following questions about properties of arithmetic of real numbers. (a) Let $A=\{1,2,3, \ldots n\}, n \in \mathbb{Z}^{+}$, and $\pi: A \rightarrow A$ be an automorphism on $A$ (an automorphism is a bijection of a set onto itself.) (b) Consider $\sum_{i=1}^{n} a_{i}$. We often use the property of commutativity $a+b=b+a$ without even thinking about it when computing sums. Since addition is a binary operation, we can only add two numbers at a time. Claim: \[ \sum_{i=1}^{n} a_{i}=\sum_{i=1}^{n} a_{\pi(i)} \] (c) State and prove a claim similar to the claim in part $2 \mathrm{~b}$ about multiplication of real numbers.

Step 1 ：Let $a_{1}, a_{2}, \ldots, a_{n} \in \mathbb{R}$ be a sequence of real numbers.

Step 2 ：Let $A=\{1,2,3, \ldots n\}, n \in \mathbb{Z}^{+}$, and $\pi: A \rightarrow A$ be an automorphism on $A$ (an automorphism is a bijection of a set onto itself).

Step 3 ：Consider the sum of the sequence, $\sum_{i=1}^{n} a_{i}$. Since addition is a binary operation, we can only add two numbers at a time.

Step 4 ：We claim that the sum of the sequence remains the same even if the order of the numbers in the sequence is changed. In other words, $\sum_{i=1}^{n} a_{i}=\sum_{i=1}^{n} a_{\pi(i)}$ for any permutation $\pi$ of the sequence $a_{1}, a_{2}, \ldots, a_{n}$.

Step 5 ：This claim is a direct consequence of the commutative property of addition, which states that the order of numbers does not affect their sum.

Step 6 ：\(\boxed{\text{Final Answer: } \sum_{i=1}^{n} a_{i} = \sum_{i=1}^{n} a_{\pi(i)}}\)