Imagine your-self in front of a class with very good undergraduates
who plan to do mathematics (professionally) in the future.
You have 30 minutes after that you do not see these students again.
You need to present a **theorem which will be 100% useful** for them.

What would you do?

**One theorem per answer please.** Try to be realistic.

For example: 30 min is more than enough to introduce metric spaces, prove existence of partition of unity, and explain how it can be used later.

**P.S.** Many of you criticized the vague formulation of the question. I agree. I was trying to make it short --- I do not read the questions if they are longer than half a page. Still I think it is a good approximation to what I really wanted to ask. Here is **an other formulation** of the same question, but it might be even more vague.

Before I liked *jewelry-type* theorems; those I can put in my pocket and look at it when I want to.
Now I like *tool-type* theorems; those which can be used to dig a hole or build a wall.
It turns out that there are jewelry-type and tool-type theorems at the same time.
I know a few and I want to know more.

to anyoneand as a result get them to understand the significance of anything. $\endgroup$ – Mariano Suárez-Álvarez Apr 4 '11 at 17:10