Understanding Bounded Intervals to Suprema and Infima

[Icebreaker]

"If I and J are bounded, then I\capJ 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 I \cap J. Similarly for the lower bound.