Well-Define of Summation of Real Number : Cauchy/Quotient Set Approach Definition of $\Bbb R$
For two sequences $\alpha,\beta : \Bbb N \rightarrow \Bbb Q$ 
Define $\alpha \sim \beta$ when $\forall e\;\;$one can pick $N$ 
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 element of Quotient set $\Bbb R = \Bbb F/\sim$, 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 
my exercise book requires me to prove whether the $[\alpha]+[\beta] = [\alpha+\beta]$ is well-defined.
I understand the "Well-define" is try to find out another equivalence class to $[\alpha]$ and $[\beta]$ such as $[\alpha] \sim [\alpha]'$ and $[\beta] \sim [\beta]'$ then whether the definition still works, but frankly my understanding doesn't reach to 'what is the meaning of definition still works?' or 'what exactly in which verification' my textbook requires me. 
Any guidance would be grateful.  
 A: $\alpha, \alpha', \beta, \beta' : \Bbb N \rightarrow \Bbb Q$
Let $\alpha\sim\alpha'$ and $\beta\sim\beta'$ then 
$\forall e \in \Bbb Q>0, \;\; \exists N_1\; s.t. \; \forall i\ge N_1\; \rvert\alpha(i) - \alpha'(i)\lvert \lt {e \over 2}\;$ and
$\forall e\in \Bbb Q>0, \;\; \exists N_2\; s.t. \; \forall i\ge N_2\; \rvert\beta(i) - \beta'(i)\lvert \lt {e \over 2}\;$
Now let   
$\;\gamma(i) := \alpha(i) + \beta(i) $
$\;\gamma'(i) := \alpha'(i) + \beta'(i)$
$\;N = sup(N_1, N_2)$
then 
$\; \forall e \in \Bbb Q>0, \;\; \exists N\; s.t.\; |\gamma(i) - \gamma'(i)|= |\alpha(i) - \alpha'(i)+ \beta(i)  - \beta'(i)| \le\; \rvert\alpha(i) - \alpha'(i)\lvert + \rvert\beta(i) - \beta'(i)\lvert \lt e$
Thus $[\gamma] = [\gamma']$ since $\gamma \sim \gamma'$ 
A: Hint: let $\alpha\sim\alpha'$ and $\beta\sim\beta'$, and define
$$
\gamma(i) := \alpha(i) + \beta(i),
\qquad
\gamma'(i) := \alpha'(i) + \beta'(i).
$$
You have to show that $[\gamma] = [\gamma']$, i.e., for every $\epsilon > 0$ there exists $N\in\mathbb{N}$ such that $|\gamma(i) - \gamma'(i)| < \epsilon$ for every $i\geq N$.
Can you estimate the difference $|\gamma(i) - \gamma'(i)|$ in terms of the differences $|\alpha(i) - \alpha'(i)|$ and $|\beta(i) - \beta'(i)|$?
