# Thread:

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

3. ## Re: Question on intervals

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

To prove "If a is less than or equal to b, then (-infinity, a) is a subset of (-infinity, b)":
If $p\in (-\infty, a)$ then p< a. Since $a\le b$ and "<" is transitive, $p<a\le b$ so $p\in (-\infty, b)$.
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