Understanding Bounded Intervals to Suprema and Infima

[Icebreaker]

"If I and J are bounded, then I[tex]\cap[/tex]J is also bounded."

Now, I was able to do this using the definition of suprema and infima and so fourth, but it is one godawful mess. I could sumbit it as is, but I was wondering if there's an easier way.

 

[Muzza]

Why would you involve suprema and infima?

Let C be an upper bound of I, and D an upper bound of J. Then max{C, D} is an upper bound of [tex]I \cap J[/tex]. Similarly for the lower bound.

 

[quicknote]

I understand the importance of clear and concise explanations in scientific literature. In regards to your statement about bounded intervals, it is important to note that the intersection of two bounded intervals will also be a bounded interval. This can be easily understood by considering the definition of a bounded interval, which is a range of values between a lower bound and an upper bound.

When two bounded intervals, I and J, intersect, the resulting interval will have a lower bound that is the maximum of the lower bounds of I and J, and an upper bound that is the minimum of the upper bounds of I and J. This ensures that all values within the resulting interval fall within the bounds of both I and J, making it a bounded interval.

In summary, the intersection of two bounded intervals will always result in a bounded interval. This can be easily understood by considering the definition of bounded intervals and the properties of suprema and infima. I hope this explanation provides a clearer understanding of bounded intervals for you.