# Math Help - Interval Proof

1. ## Interval Proof

We want to prove that $(a, \infty) \subseteq [b, \infty) \Leftrightarrow a \geq b$.

Proof: $x \in (a, \infty) \Rightarrow x \in [b, \infty) \Rightarrow a \geq b$.

$a \geq b \Rightarrow x \in (a, \infty) \Rightarrow x \in [b, \infty) \Rightarrow (a, \infty) \subseteq [b, \infty)$.

Does this look right? I drew a number line and used 'intuitive reasoning' to show that it is true.

Thanks

2. Originally Posted by shilz222
We want to prove that $(a, \infty) \subseteq [b, \infty) \Leftrightarrow a \geq b$.
Forward.
If $x\in (a,\infty)$ then $x\in [b,\infty)$ thus, $a.
Argue by contradiction, say $a then choosing $x=(a+b)/2$. This leads to contradiction, why?

Converse.
If $x\in (a,\infty)$ then $a, but then $b\leq a < x$. Thus $x\in [b,\infty)$. Thus $(a,\infty)\subseteq [b,\infty)$.

3. contradicts the fact that $b \leq \frac{a+b}{2}$.