Definition of $\Bbb R$
For two sequences $\alpha,\beta : \Bbb N \rightarrow \Bbb Q$
Define $\alpha \sim \beta$ when $\forall e\;\;$ there $\exists N$ s.t. $\forall i \ge N$, $\lvert \alpha(i)-\beta(i)\rvert \lt e$
Since $\sim$ is equivalence relation of the set $\Bbb F = \{\alpha: \Bbb N \rightarrow \Bbb Q$},
call the each element of quotient set, $\;\;F/\sim\;\;$, real number.
Definition of Summation among Real Numbers
Define $[\alpha]+[\beta] = [\alpha+\beta]$ (*)
$\alpha + \beta$ is defined as $i \mapsto \alpha(i) + \beta(i)$
Question
I had already proved that above definition of summation is well-defined.
Now I want to prove whether this operation holds the associative property.
Proof
for $\alpha, \beta, \gamma : \Bbb N \rightarrow \Bbb Q$,
we need to check below two conditions:
1) $([\alpha] + [\beta]) + [\gamma] = [\alpha] + [\beta] + [\gamma]\;\;$ and
2) $[\alpha] + ([\beta] + [\gamma]) = [\alpha] + [\beta] + [\gamma]$
for 1), since $[\alpha]+[\beta] = [\alpha+\beta]$,
LHS of 1) = $[\alpha+\beta] + [\gamma] $ and
$[\alpha+\beta] + [\gamma] =[\alpha+\beta+\gamma]$ by defintion of (*)
for 2), since$\;\; [\beta]+[\gamma] = [\beta + \gamma]$,
LHS of 2) = $[\alpha] + [\beta + \gamma]$ and
$[\alpha] + [\beta + \gamma] = [\alpha + \beta + \gamma] $ by definition of (*)
Question
I am not familiar to logical thinking.
Is above proof is sufficient or logically clear under the notion of checking the associative property?
No, you don't have to check your 1) and 2). Note that $[\alpha]+[\beta]+[\gamma]$ is undefined at the moment. Instead you have to check that $$\bigl([\alpha]+[\beta]\bigr)+[\gamma]=[\alpha]+\bigl([\beta]+[\gamma]\bigr)\ .\tag{1}$$ For the proof use that $[\alpha]+[\beta]$ is represented by $\alpha+\beta$, hence $$\bigl([\alpha]+[\beta]\bigr)+[\gamma]=[\alpha+\beta]+[\gamma]=[\alpha+\beta+\gamma]\ .$$ Similarly for the RHS of $(1)$.