Theorem:
Let $A$ be a set. For a cover $C$ of $A$, let
$$D=\{I_n + x: I_n \in C, x\in\Bbb R\}$$
Prove that $D$ covers $A+x$.
Proof:
Given:
- Two sets $A$ and $A+x$.
- $C$ is a cover for $A$. That is $C=\{I_n|A\subseteq UI_n\}$
- $D=\{I_n+x|I_n\in C\}$
To prove:$D$ covers $A+x$.
That is, to prove $A+x \subseteq \cup(I_n+x)$.
For that let $a_1+x\in A+x$ for some $a_1\in A$
$a_1\in A\;\implies a_1 \in \cup I_n\implies (a_1+x)\in \cup I_n+x$.
That is $A+x \subseteq \cup(I_n+x)$.
Hence $D$ covers $A+x$.
I am trying to learn proof writing.
This is one of my proofs, but my friends suggested that instead of using "for some " in the line "For that let $a_1+x\in A+x$ for some $a_1 \in A$" we should use "for all".I googled about quantifiers, read through the Wikipedia articles, but still, I am confused that why should I change that wording.
If $p \in A+x$, then this means that $p = a + x$ for some $a\in A$. Your proof looks fine to me.