This is my first post on here so let me know if I can do anything better. I'm currently beginning to study a bit of real analysis. I've come across some proofs involving supremum which are proofing the same question but in a slightly different way. I was mainly wondering which has the best style and is recommended for someone learning analysis and for someone reading.
Question:
Let $ A\subset \mathbb{R}$ be bounded above and let $c\in \mathbb{R}$. Define the set $c+A := \{c+a:a\in A\}$. Prove that $\sup(c+A) = c+\sup(A)$.
Proof 1:
Since $A$ is bounded above, the completeness axiom tells us $\sup(A)$ exists. By definition, for all $a\in A, a\leq \sup(A)$. It follows that for all $a\in A, c+a\leq c+\sup(A)$. Therefore, c+sup(A) is an upper bound for $c+A$. Since $\sup(c+A)$ is the least upper bound for $c+A$, $\sup(c+A)\leq c+\sup(A).$ Additionally, for all $a\in A, a\leq \sup(A) \Rightarrow \forall a\in A, a\leq \sup(c+A)-c$. Therefore, $\sup(c+A)-c$ is an upper bound for A. Since $\sup(A)$ is the least upper bound for A, $\sup(A) \leq \sup(c+A)-c \Rightarrow c+\sup(A) \leq \sup(c+A)$. So, it follows that $\sup(c+A) = c+\sup(A)$.
Proof 2:
- Let $x\in c+A$. By definition of the set, $x=c+a$ where $a\in A$. Since $A$ is bounded above, the completeness axiom tells us $\sup(A)$ exists. By definiton of upper bound, $a \leq \sup(A)$. So, $x= c+a \leq c+\sup(A)$. Since $x$ was arbitrary, $\forall x\in c+A, x\leq c+\sup(A)$.Therefore, c+sup(A) is an upper bound for $c+A$. Since $\sup(c+A)$ is the least upper bound for $c+A$, $\sup(c+A)\leq c+\sup(A).$
- Let $a\in A$. Then $c+a\in c+A$. By definiton of $\sup(c+A), c+a\leq \sup(c+A) \Rightarrow a\leq \sup(c+A)-c$. Since $a$ was arbitrary, $\forall a\in A, a\leq \sup(c+A)-c$. Therefore, $\sup(c+A)-c$ is an upper bound for A. Since $\sup(A)$ is the least upper bound for A, $\sup(A) \leq \sup(c+A)-c \Rightarrow c+\sup(A) \leq \sup(c+A)$.
- Together, it follows that $\sup(c+A) = c+\sup(A)$.
I feel like I'm drawn to the second proof more since it follows the rules of proofs that I've learned in my previous proof class such as proving for all statements. But, are both ways valid ways of proving the statements and which one is recommended for real analysis and future learning? Also, any other tips would be amazing! Thanks in advance!