Re: Question on intervals

Assume $\displaystyle a \leq b $ then if a number $\displaystyle p \in (-\infty, a) $ then $\displaystyle p < a $, so $\displaystyle p < a \leq b $ thus $\displaystyle p < b $ so $\displaystyle p \in (-\infty, b) $

Re: Question on intervals

The standard way to prove "A is a subset of B" is to start "if $\displaystyle x\in A$, then use the definitions of A and B to conclude "$\displaystyle x\in B$".

To prove "If a is less than or equal to b, then (-infinity, a) is a subset of (-infinity, b)":

If $\displaystyle p\in (-\infty, a)$ then p< a. Since $\displaystyle a\le b$ and "<" is transitive, $\displaystyle p<a\le b$ so $\displaystyle p\in (-\infty, b)$.