Prove this completely obvious question

  • Thread starter AGNuke
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I think it's a great book, but it's not for everyone, and it's not a book for someone who's new to the subject.In summary, when it comes to "obvious" questions in mathematics, it's important to understand definitions and to have a lot of practice with different methods of proof. Additionally, don't be discouraged if you have trouble proving things that seem "obvious" - it's all part of the learning process.
  • #1
AGNuke
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The Question -

Let ##\mathbb{A}## be a non-empty subset of ##\mathbb{R}## and ##\alpha \in \mathbb{R}##. Show that ##\alpha = sup\mathbb{A}## if and only if ##\alpha - \frac{1}{n}## is not an upper bound of ##\mathbb{A}## but ##\alpha + \frac{1}{n}## is an upper bound of ##\mathbb{A}\;\forall \;n \in \mathbb{N}##.

My solution -
I tried to approach this completely obvious question (as per to me) with two contradictions - one for each case.

Let ##\alpha## be an upper bound but ##\alpha + \frac{1}{n}## isn't upper bound. Then, ##\exists \; a\in \mathbb{A}## such that
$$\alpha + \frac{1}{n} \leq a \Rightarrow a - \alpha \geq \frac{1}{n}\; \forall \;n \in \mathbb{N}$$
but from the first condition,
$$\alpha \geq a \;\forall\;a\in\mathbb{A} \Rightarrow \Leftarrow $$
Hence, our assumption is flawed. Hence ##\alpha - \frac{1}{n}## must be upper bound if ##\alpha## is to be the supremum of ##\mathbb{A}##.

Similarly for the second condition (by contradiction)

Help required -
The thing is, I simply can't get to solve such "simple" and "obvious" question in such methods without toiling for hours. If possible, can someone refer me to course materials where these "obvious" questions are solved by such "not-so-obvious" methods? I am really troubled and unable to understand things like these in class, furthermore, my exams are drawing near.
 
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  • #2
"Obvious" questions in mathematics often depend on definitions - and definitions often come from a previous result, such as a theorem which proves the existence of some "thing" ... if that thing is useful it has a ready-made definition.

So you need to know the definitions and understand how they originated. This (hopefully) provides a more intuitive understanding ... and makes many things obvious.

Lots of practice makes things more obvious ...
 
  • #3
Practice is fine, but is there any reference course material where I can collectively see such theorems (like Archimedes Principle, Bolzano-Wierstrass Theorem, Cauchy criterion, etc.) with ease. I have a hard time to understand them from pages like on wikipedia.
 
  • #4
For some good advice try Terrance Tao:
http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/

If you are looking for a general reference work with lots of stuff in it - try any of the books especially written for "mathematical methods for physics" ... they include lots of topics, and usually have good references.
 
  • #5
Keep in mind that in addition to learning mathematics, you're also (likely) learning how to prove things as well as write those proofs. It's not uncommon for students to be asked to prove "obvious" statements so that they can practice various methods of proof.

In this case, you've really only done about 25% of the work that needs to be done. The statement that you're being asked to prove is an "if and only if" statement, and you've only demonstrated part of one of those implications. Furthermore, there is an incorrect (though easily fixable) statement as well as a typo in the part of the proof that you have done.

It's not always the case that statements which are intuitively "obvious" are easy to prove or even true. The point of doing rigorous mathematics is to be, well, rigorous.

I use Rudin's Principles of Mathematical Analysis as a reference for topics like the ones that you have mentioned.
 

1. What do you mean by "prove this completely obvious question"?

When we say "prove this completely obvious question", we are referring to providing evidence or logical reasoning to support a statement or claim that may seem self-evident or common sense to some people.

2. Why would someone ask for proof of something that is obvious?

There could be various reasons for this. One possibility is that the person asking for proof may not find the statement or claim to be as obvious as others do and therefore wants more convincing evidence. Another reason could be for the sake of thoroughness and avoiding assumptions.

3. How can you prove something that is obvious?

Proving something that is obvious can be done through the use of logical reasoning, empirical evidence, or by providing examples and analogies. It may also involve breaking down a complex concept into simpler, more easily understood components to make the proof more clear.

4. Is it necessary to prove something that is obvious?

It depends on the context and the audience. In some cases, providing proof may not be necessary if the statement or claim is widely accepted and understood. However, in certain scientific or academic settings, it is important to provide evidence and support for all claims and statements, regardless of how obvious they may seem.

5. Can something be proven to be completely obvious?

Yes, something can be proven to be completely obvious. This is usually done by using a combination of logical reasoning, empirical evidence, and clear explanations to support the statement or claim. However, what may seem completely obvious to one person may not be as obvious to another, so it is important to consider different perspectives when attempting to prove something as completely obvious.

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