1. ## Question on intervals

My question is:

i) Is the following statement true or false, and explain why: "If a is less than or equal to b, then (-infinity, a) is a subset of (-infinity, b)".

I drew a diagram which showed that this was true, but how would you go about proving it formally?

Any help would be appreciated

2. ## 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 $\displaystyle p \in (-\infty, b)$

3. ## 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 $\displaystyle p\in (-\infty, b)$.