## From minimum to infimum: Math is just a logical game

The true Math is a continuous exercise in logic. A good teacher makes that logic visible and tangible. A genius teacher does the logic mentally and only displays the result. I am trying to be as good a teacher as humanly possible.

## Steps to develop a new definition

**Step 1**. Start with a simple example.

What is the minimum of the set ? In my class everybody says .

**Step 2**. Give the most visual definition.

Here is a good candidate: the minimum of a set is **its leftmost point**.

**Step 3**. Formalize the definition you gave.

We said "its point". This means should be an element of . "Leftmost" is formalized as for any . Thus the **formal definition** should be: the number is called a **minimum** of a set if

1) for any and

2) .

**Step 4**. Look at bad cases when the definition fails. Try to come up with a generalized definition that would cover the bad cases.

## Here is a bad case: meet the infinity

Let . In my class, some students suggested and . Can you see why neither is good? See the explanation below if you can't. I would try . It still satisfies part 1) of the above definition but does not belong to . This is the place to stretch one's imagination.

**Statement 1**. No element of satisfies the formal definition above.

**Proof**. Take any element . Then it is larger than 1 and the number is halfway between and 1. We have shown that to the left of any there is another element of . So no element of is leftmost.

In the game of chess, the one who thinks several moves ahead wins. A stronger version of Statement 1 is the following.

**Statement 2**. To the left of any there are infinitely many elements of .

**Proof**. Indeed, set The numbers approach 1 from the right. As increases, they become closer and closer to 1. For any given , infinitely many of these numbers will satisfy .

**Step 5**. When thinking about a new definition, try to exclude undesirable outcomes.

Since for condition 2) is not satisfied, one might want to omit it altogether. However, leaving only part 1) would give a bad result. For example, would satisfy such a "definition", and it would be bad for two reasons. Firstly, there is no uniqueness. We could take any number . Secondly, if you take any , in the interval there are no elements of , so there is no reason to call such a number a minimum of .

**Definition**. A number is called an **infimum** of (denoted ) if

I) for any and

II) in , there is a sequence approaching .

The above discussion shows that .

**Exercise 1**. Repeat all of the above for . First define the maximum of a set. Then modify the definition to obtain what is called a **supremum**.

**Exercise 2**. Let , where is the basis of the natural log and is the ratio Length of circumference/Diameter of that circumference. Find the infimum and supremum of . Simply naming them is not enough; you have to prove that both parts of the definitions are satisfied.

**Exercise 3**. The infimum is also defined as the largest lower bound. Can you prove equivalence of the two definitions?

**Exercise 4**. The supremum is also defined as the least upper bound. Can you prove equivalence of the two definitions?

**Exercise 5**. Consider any two sets and their union. What is the relationship between the infimums of the three sets?

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