Request of a proof concerning real sequences

Question

Prove the following:

For all $a \in \mathbb{R}$, for all $t >0$, there exists a $w \in \mathbb{Q}$ such that $|w-a|

Sorry if it is unclear.

Please ask for what is unclear before downvote.
I'll try to explain to my best.

Answer 1

Let $a \in \mathbb{R}$ and $t>0$. We wish to find $w \in \mathbb{Q}\cap(a-t,a+t)$. Choose $N\in\mathbb{N}$ such that $\frac{1}{N}<2t$ and let's define $M:=\{ \frac{n}{N} : n \in\mathbb{N} \}\subseteq\mathbb{Q}$. Then $M\cap(a-t,a+t)\neq\emptyset$. If this were false, we take the biggest $m_1\in\mathbb{N}$ such that $\frac{m_1}{N}a+t$. This leads to $2t<\frac{m_1+1}{N}-\frac{m_1}{N}=\frac{1}{N}<2t$ and we have a contradiction. Thus $M\cap(a-t,a+t)\neq\emptyset$ and we're finished. We've found that $\forall a\in\mathbb{R}$ and $\forall t>0$ there exists a $w\in\mathbb{Q}$ such that $|w-a|