# Help Needed: Struggling to Get Started with Physics

• mynameisfunk
In summary, the problem asks for an axiom that tells when a sequence of real numbers converges. There are three parts to this problem, each with its own explanation. The first part asks for an axiom that tells when a sequence of reals converges. The second part asks for an axiom that tells when a sequence of nested, bounded, closed intervals contains at least one common point. The third part asks for an axiom that tells when a sequence of reals converges to a particular real number.
mynameisfunk
https://www.physicsforums.com/attachments/28380Have really no idea where to start with this one. Anyone want to help me get started?

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well that is about as complicated a problem i have seen, to bash out a simple idea.

basically it says to try 1/2. if that doesn't work then you try next either 1/4 or 3/4, and more precisely you try 1/4 if 1/2 was too big, and you try 3/4 if 1/2 was too small.

say 1/2 was too small and you next try 3/4. then eiother it works or it doesn't.

if 3/4 is too small you try next 7/8, and if 3/4 is too big try next 5/8.get it? eventually (after the end of the world that is) you have an infinite number of tries x all getting nearer each other and also ll values x^n getting nearer to y.
so you work to prove that the x's converge to something whose mth power must converge to y.

for this you need an axiom that tells yiou when a sequence of =reals converges.

or as this problem puts it, you need an axiom that says a shrinking sequence of nested, bounded, closed intervals contains at least one common point, and exactly one point if they shrink to zero in length.

or maybe it is rigged to use the fact that a bounded monotone sequence of reals converges.
.

to do the problem as asked just put your nose down and try to follow the steps one at a time blindly.

i myself dislike such problems. i mean what do you learn from them? maybe a little technical skill.

THANK YOU. You have made that much clearer. Tomorrow I will work on it and post my work.

OK, I am confused, the first 3 parts seem trivial and i can't imagine doing anything but going through the process a few times and showing that they are all true by induction...
Part (d) however, i do not see how this is true. How am I to know how far away an and bn are from each other? I would have thought they could be anything. I specifically don't see how/why |a1-b1|=1/2

well without looking back i recall the next numbers ere the average of the previous ones so they seem to get closer by one half each time.

mathwonk said:
eventually (after the end of the world that is) you have an infinite number of tries x all getting nearer each other and also ll values x^n getting nearer to y.
so you work to prove that the x's converge to something whose mth power must converge to y.

for this you need an axiom that tells yiou when a sequence of =reals converges.

or as this problem puts it, you need an axiom that says a shrinking sequence of nested, bounded, closed intervals contains at least one common point, and exactly one point if they shrink to zero in length.

or maybe it is rigged to use the fact that a bounded monotone sequence of reals converges.

Would this theorem be the Bolzano-Weierstrass Theorem??(Every bounded infinite subset of R^k has a limit point in R^k). Specifically for part (e).

mynameisfunk said:
|a1-b1|=1/2

You start with b0=1 and a0=0. Then we check to see if $$\left( \frac{1}{2}^n\right)=y$$. If so, we're done. If not, there are two possibilities: we make a1=1/2 or we make b1=1/2, and leave the other number alone depending on whether $$\left(\frac{1}{2}\right)^n$$ is too big or too small.

Then |a1-b1| is either |0-1/2| or |1/2-1|

I would like to post my work on this and see if anybody has a problem. I am very unsure on how to do this...
(a)initially, an$$\leq$$bn and an$$\leq$$c$$\leq$$bn. If cm>y we set an=an-1 and bn=c so that we still have an-1$$\leq$$an$$\leq$$bn$$\leq$$bn-1. If cm<y then we we set an=c and bn=bn-1 so that we still have an-1$$\leq$$an$$\leq$$bn$$\leq$$bn-1 and the sequence satisfies part (a).

(b)Since all bn and an are reals, they are ordered. Since a0$$\leq$$a1$$\leq$$...$$\leq$$an$$\leq$$bn$$\leq$$...$$\leq$$b1$$\leq$$b0, then [an,bn]$$\subseteq$$[an-k,bn-k] for any positive integer k$$\leq$$n.

(c) Note that an<c and bn>c. Since in our process, if cm$$\neq$$y, we shift our nested intervals(infinitely many times if necessary) so that y1/m is contained in all of them, then the bounds of these sequences, namely an and bn for all n satisfy an<y1/m<bn$$\Rightarrow$$ amn<y<bmn for all n.

(d)Our process of setting either bn=c or an=c depending on whether ym>c or ym<c makes our interval smaller by 1/2 each time, where initially, the interval is of length 1 ( |0-1| ). Hence |bn-an|=2-n

(e) I would like to just use the Bolzano Weierstrass Theorem here... This is as far as I have done..

There are two parts to point e: First is that the intersection is non-empty, and the second is that there is only one point. You can use a convergence argument to get that it's not empty: the sequence an is bounded above and increasing, so converges to something we'll call a; the sequence bn is bounded below and decreasing, so has a limit also called b. By preservation of weak inequalities, $$a\leq b$$ (make sure this is true! otherwise you're stuck in the mud)

Prove that [a,b] is in the infinite intersection, and use that to prove that a=b also

I think the buzzword for this kind is the bisection method.

## 1. What is physics?

Physics is the scientific study of matter, energy, and their interactions. It seeks to understand the fundamental laws and principles that govern the behavior of the physical world.

## 2. Why is understanding physics important?

Physics helps us understand the natural world around us, from the smallest particles to the largest galaxies. It also has many practical applications, such as developing new technologies and solving real-world problems.

## 3. What are some key concepts in physics?

Some key concepts in physics include motion, energy, forces, and laws of motion. Other important topics include thermodynamics, electromagnetism, and quantum mechanics.

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## 5. What are some tips for understanding physics concepts?

Try to break down complex concepts into smaller, more manageable parts. Use visual aids, such as diagrams or graphs, to help you understand relationships and visualize abstract concepts. Practice solving problems and seek help from tutors or classmates when needed.

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