Question

Help with "proof writing". I started taking a mathematics course, and only realized too late that the calculus course is a higher level requiring proof writing. For example prove that the limit as x approaches x of x*sinx exists or does not exist. I have no background in proof writing, so if anyone can help me out here, I would more than appreciate it. Thanks. Rafi

Answer 1

As x approaches 0*

Answer 2

The first step to a lot of proof writing is understanding what's being asked. What is the definition of a limit?

Answer 3

you should know what rout you will take also, direct proof, contraposition, or contradiction.

Answer 4

zz0ck3r-I beleive the proof writing needed for the course is all of the above.
Jemuraray- I understand the concepts, but don't know the correct format of proof writing. Is it something complicated or fairly understandable?

Answer 5

ok lets do the proof for "all evennumbers square are even numbers"
assume x is an element of Z
let x be even. So x = 2k for some k in Z
so x^2 = (2k)^2 = 4k^2 = 2(2k^2).
Since 2k^2 is an integer 2(2k^2) is even.
thus every even number squared is even.

Answer 6

lets proof with contradicton
so negate the whole thing
Assume x is even and x^2 is odd.
same steps....
Thus a contradiction is x^2 is even and x^2 is odd....thus...

Answer 7

contrapostion
assume x^2 is odd and x is odd
let x E N
Let x = 2k+1.
thus x^2 = (2k+1)2 = 4k^2 + 4k + 1 = 2(2k^2+2k) + 1 thos x^2 is odd. so....

Answer 8

There isn't necessarily a single format that you must use to prove things. The general idea is that you explain what you will be proving, show what assumptions are being made, and then logically trace a path to your conclusion. A rough outline for a limit proof is

We seek to prove that the function defined by f(x) = x sin(x) has a limit L = 0 as x -> 0. We must therefore show that we can make |f(x) - L| as small as we want by making x arbitrarily small. *Then proceed to show this*

Answer 9

You may be able in your situation to use calculus or your teacher may want you to prove anything you use... It all depends on what your doing.
exp
show if a linear function is increasing then it is always increasing. You could do this with calc or you could assume x>y and show that f(x) > f(y).... so it all depends.

Answer 10

Thanks for your examples. I can follow it by immitation, but don't know if I can come up with those proofs so quickly and naturally. What would be considered a "normal" time to get used to proof writing?

Answer 11

I'm going to show you the full, unabridged definition of a limit, and you tell me if it is more complicated than what you need, and if so, how much.

\[ \text{ We say that } \lim_{x\rightarrow a} f(x) = L\text{ if, given } \epsilon >0, \text{ we can find some } \delta >0 \text{ such that }|x-a| < \delta\]
\[\text{ implies that } |f(x)-L| < \epsilon.\]

Answer 12

oops. That line should end with
\[ \text{ such that } |x-a|< \delta\]

Answer 13

That's the definition I have according to the material. Using epsilon and delta to find the range around the lmit etc.

Answer 14

I will show you the standard limit proof, and if you require further elaboration let me know:

Answer 15

We seek to show that
\[\lim_{x\rightarrow 0} x\cdot \sin(x) = 0\]
\[ \text{ let } \epsilon >0 \text{ be given. We seek some } \delta > 0 \text{ such that }\]
\[|x-0|<\delta \text{ implies that } |x\cdot \sin(x)| < \epsilon. \]
\[ \text{ We note that } |\sin(x)| \leq 1 \text{ for all x. Therefore, } |x\cdot \sin(x)| \leq |x| \text{ for all x.}\]
\[\text{ Therefore, simply let }\delta = \epsilon. \text{ Then, }|x|<\delta = \epsilon \text{ implies that } |x\cdot \sin(x)| \leq |x| < \epsilon. \]
Thus the proof is concluded.

Answer 16

Thank you very much. I am starting to make heads and tails of it now that you broke it down to its elements. Is this considered a hard question relatively or beginner's level?

Answer 17

In terms of proofs, this one is fairly obvious. However, if it's your first experience with them, then they won't get too impossible. The only potentially tricky part is the algebraic manipulation within the proof, but it's typically fairly straightforward.

Answer 18

Alright, well thanks for your help:) You really put things into perspective. I'll see how I manage now.