Heine–Borel theorem
The Heine-Borel Theorem is a rule we use in math to help us know when some really important things happen. Basically, it says that if we have a bunch of points that we draw on a line, and we want to cover them all up with some little shapes that are all the same size, then we can always do that!
Here's an example: imagine we have a line with a whole bunch of dots on it, like this:
......o...o..o.........o....o........o............o..o.o
And we want to cover all these dots up with little circles that are all the same size. The Heine-Borel Theorem is like a magic rule that helps us know if we can do that or not.
Basically, the rule says that if we can draw a big square or rectangle that contains all of our dots, then we can always find some little circles that will cover up all the dots inside our big square or rectangle. And not only that, but the circles will all be the same size!
So if we look at our line of dots, we can imagine drawing a big square around all of them, like this:
+----------------------+
|......o...o..o.........o....o........o............o..o.o|
| |
| |
+----------------------+
Once we have this big square, we can be sure that we can always cover up all the dots with some circles that are all the same size. It's like magic!
So that's the Heine-Borel Theorem. It's a rule that helps us know when we can cover up all the points on a line with little shapes that are all the same size. And the rule says that if we can draw a big square or rectangle that contains all the points, then we can always do it!
Here's an example: imagine we have a line with a whole bunch of dots on it, like this:
......o...o..o.........o....o........o............o..o.o
And we want to cover all these dots up with little circles that are all the same size. The Heine-Borel Theorem is like a magic rule that helps us know if we can do that or not.
Basically, the rule says that if we can draw a big square or rectangle that contains all of our dots, then we can always find some little circles that will cover up all the dots inside our big square or rectangle. And not only that, but the circles will all be the same size!
So if we look at our line of dots, we can imagine drawing a big square around all of them, like this:
+----------------------+
|......o...o..o.........o....o........o............o..o.o|
| |
| |
+----------------------+
Once we have this big square, we can be sure that we can always cover up all the dots with some circles that are all the same size. It's like magic!
So that's the Heine-Borel Theorem. It's a rule that helps us know when we can cover up all the points on a line with little shapes that are all the same size. And the rule says that if we can draw a big square or rectangle that contains all the points, then we can always do it!