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"Prove that the sum of two rational numbers is a rational number."

I just started on proof writing, so I'll just like to verify if I'm not missing anything here, and get some comments about the style.

The attempt at a solution

Theorem.If [tex]a,b \in \mathbb{Q}[/tex] then [tex]a+b \in \mathbb{Q}[/tex]

Proof.Given [tex]a,b \in \mathbb{Q}[/tex]

we have some [tex]p_1,p_2,q_1,q_2 \in \mathbb{Z}[/tex] satisfying

[tex]a=\dfrac{p_1}{q_1} \, b=\dfrac{p_2}{q_2}[/tex]

which implies that

[tex]a+b

=\dfrac{p_1}{q_1}+\dfrac{p_2}{q_2}

=\dfrac{p_{1}q_{2}+p_{2}q_{1}}{q_{1}q_{2}}[/tex]

By closure of [tex]\mathbb{Z}[/tex] under addition and multiplication, it follows from the assumption that [tex]p_1,p_2,q_1,q_2 \in \mathbb{Z}[/tex] that

[tex]p_{1}q_{2}+p_{2}q_{1}[/tex] and [tex]q_{1}q_{2}[/tex] are also integers.

Hence, [tex]a+b[/tex] can be expressed as the ratio of 2 integers. By definition, [tex]a+b[/tex] is rational. [tex]\square[/tex]

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# Homework Help: Is this correct? (sum of two rational numbers is rational)

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