# Proof of Sup & Inf Homework Statement

• flipsvibe
In summary: Most of the time, a symbol will correspond to a numerical value. However, there are a few cases where a symbol doesn't correspond to a numerical value, and you will have to use a different notation. One of these cases is when a symbol represents an entire expression rather than just a numerical value. For example, the symbol "*" can be used to stand for multiplication, even when there are other symbols in the expression that represent other operations. In summary, using mathematical notation can make your writing clearer and easier to read.
flipsvibe

## Homework Statement

Let a(n) and b(n), n$$\in$$N, be some real numbers with absolute value at most 1000. Let A={a(n), n$$\in$$N}, B={b(n), n$$\in$$N}, C={a(n) + b(n), n$$\in$$N}. Show that

inf A + sup B $$\leq$$ sup C $$\leq$$ sup A + sup B

## The Attempt at a Solution

I was thinking that I could show that inf A + sup B = 0, and that sup C is larger than 0, and then that sup C = sup A + sup B. The only problem is that I am terrible at writing formal proofs, and could really use some help with the language, and (probably) my logic.

No; you have no idea what $$a(n)$$ and $$b(n)$$ are, so you cannot say anything about the exact values of $$\inf A + \sup B$$, et cetera.

The definition of least upper bound has two parts. $$\alpha$$ is the least upper bound of a set $$S$$ of real numbers, $$\alpha = \sup S$$, if:
1. $$\alpha$$ is an upper bound of $$S$$, that is, $$\alpha \geq s$$ for every $$s \in S$$;
2. if $$\beta$$ is also an upper bound of $$S$$, then $$\alpha \leq \beta$$.
To prove an inequality about a least upper bound, you will often use just one of these two parts, because each part constrains the least upper bound in a different direction: part 1 says that $$\alpha$$ is not too small, while part 2 says that it is not too large.

So, start from $$\sup C$$, and figure out how to use the two parts of the definition of least upper bound to constrain $$\sup C$$ on its two sides, using the other quantities mentioned.

Ok, so can I say that
C=a(n) + b(n) $$\Rightarrow$$ Sup C $$\geq$$ a(n) + b(n) $$\forall$$ n$$\in$$N?

flipsvibe said:
Ok, so can I say that
C=a(n) + b(n) $$\Rightarrow$$ Sup C $$\geq$$ a(n) + b(n) $$\forall$$ n$$\in$$N?

The left side of this implication doesn't make sense, because $$C$$ is a set. Except for that, you're right. $$C = \{ a(n) + b(n) : n \in \mathbb{N} \}$$, so every upper bound $$\gamma$$ for $$C$$ satisfies $$\gamma \geq a(n) + b(n)$$ for every $$n \in \mathbb{N}$$; in particular, $$\sup C$$ is an upper bound for $$C$$, so for every $$n \in \mathbb{N}$$, $$\sup C \geq a(n) + b(n)$$.

Two points of advice on mathematical writing:
1. You may wish to use words instead of symbols to express things like "therefore", "for all", and so on. There's nothing wrong with doing so, and it prevents you from making "grammatical errors" with your symbols. Grammatical errors with symbols are easier to make than with words, and their consequences are worse.
2. A specific example of this is that, when writing in symbols, quantifiers go before the uses of the variables they bind, not afterward. The terse and correct way to write what you intended to write is: $$C = \{ a(n) + b(n) : n \in \mathbb{N} \} \implies \forall n\in\mathbb{N}.\,\sup C\geq a(n) + b(n)$$. A critical reader might say that you wrote refers to an instance of $$n$$ that comes from outside the formula, and the quantifier at the end binds a variable $$n$$ that has no relation to the one used in the rest of the formula. Of course everyone will know what you probably meant, but it's better to be precise.

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## What is the definition of a supremum and infimum?

A supremum (or least upper bound) of a set is the smallest number that is greater than or equal to all the elements in the set. An infimum (or greatest lower bound) of a set is the largest number that is less than or equal to all the elements in the set.

## Why is it important to understand supremum and infimum?

Understanding supremum and infimum is important in mathematical analysis because they help us determine the bounds of a set and can be used to prove important properties of functions and sequences.

## How do you find the supremum and infimum of a set?

To find the supremum of a set, we can either use the definition and check if a number is greater than or equal to all the elements in the set, or we can use the supremum property, which states that the supremum of a set is the smallest number that is greater than or equal to all the elements in the set. Similarly, to find the infimum of a set, we can either use the definition and check if a number is less than or equal to all the elements in the set, or we can use the infimum property, which states that the infimum of a set is the largest number that is less than or equal to all the elements in the set.

## What is the relationship between supremum and infimum?

The supremum and infimum are related in that the supremum is always greater than or equal to the infimum. This is because the supremum is the smallest number that is greater than or equal to all the elements in the set, while the infimum is the largest number that is less than or equal to all the elements in the set.

## Can a set have multiple supremums or infimums?

Yes, a set can have multiple supremums or infimums. For example, if we have a set containing only one element, that element would be both the supremum and the infimum of the set. Also, if a set is unbounded, then it does not have a supremum or infimum.

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