1. ## Interval Proof

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

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

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

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 $\displaystyle (a, \infty) \subseteq [b, \infty) \Leftrightarrow a \geq b$.
Forward.
If $\displaystyle x\in (a,\infty)$ then $\displaystyle x\in [b,\infty)$ thus, $\displaystyle a<x \implies b\leq x$.
Argue by contradiction, say $\displaystyle a<b$ then choosing $\displaystyle x=(a+b)/2$. This leads to contradiction, why?

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

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