In his book, Euler gives the definition of a variable to be : "A variable quantity is an indeterminate or universal quantity, which includes within itself all completely determined values." What does he mean exactly in the last part of the sentence?
And if i have correctly understood the proof assumes that the bounded sequences actually have some part of their terms, let's say that an interval of the sequence could be finite and there is another which is infinite. Why is that?
No i am not it was just a theorem, but it seemed odd until now thanks it seems more clear. And about i mentioning the line; i just wanted to know how does the bisection looks like which is clear already by your explanation.
I have come across the proof of a theorem and i am unsure of some specific points in the proof so i hope someone could enlighten me. Here is the theorem and the proof straight from the book :
Theorem. Every bounded sequence possesses at least one limiting point.
Proof : We again determine the...
Haha there is a lot of small going on there.Anyways jokes aside so in this kind of proofs , i mean in general for epsilon proofs you actually do consider the epsilon you choose or give or even the epsilon itself to be a "fixed" positive number right?
Hello , i was just wondering if anyone could clarify one thing in this proof (its from Konrad Knopp book on infinite series) : If (x0,x1,...) is a null sequence, then the arithmetic means
xn'= x0+x1+x2+...+x/n+1 (n=1,2,3,...)
also forms a null sequence.
Proof: If ε >0 is given, then m can be...
Yes sorry if i pointed out my idea wrong and yes i seem to get everything since i am planning to try to go from mostly analysis into calculus other way around or in other words to build my mathematical maturity before i go into calculus thought in high school since i do not have any strong...
But do i need experience with calculus for this book to continue on with it ? Because i feel like i do understand everything and how to use them and even the proofs
I have come across this inequality:$$ g≤ log\ n <g + 1$$
We assume that the base of the log is b >1 and n is all the natural numbers. I would like to know if anyone could provide a proof regarding this and mention for what g ? Is it for all the g which are integers ?
Yes indeed i do actually refer to other sources on any detail that i find that does not appeal or i do not understand it. Just one note about the book, it is basically full of theorems: theorem after theorem sometimes maybe sometimes there examples and explanation about the definition or any...