Proof of Sup & Inf Homework Statement

[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.

 

[ystael]

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.

 

[flipsvibe]

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

 

[ystael]

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.