Thread: Inequality involving Integration

Okay, its concerened with michael spivak's book.

there is this one inequality or rather symbol i see, ever since the definition. what exactly does the denote here?? i think it stands for some small distance error, or error in distance towards the y-axis, it does not entirely define what the \epsilon stands for.

also this inequality;

,

i know what the inequality represents but, just dont get the proof provided by the book. The other proofs are alright, though still very challenging indeed.

can anyone explain this concern i have?

thank you

typically when i see an epsilon it is used to define a very small number, but it may mean something else in your book

Originally Posted by Heinsbergrelatz

Okay, its concerened with michael spivak's book.

there is this one inequality or rather symbol i see, ever since the definition. what exactly does the denote here?? i think it stands for some small distance error, or error in distance towards the y-axis, it does not entirely define what the \epsilon stands for.

also this inequality;

,

i know what the inequality represents but, just dont get the proof provided by the book. The other proofs are alright, though still very challenging indeed.

can anyone explain this concern i have?

thank you

You need to do a better job of defining your terms.

1. will be defined somewhere in the text, and is usually an arbitary positive number. You can usually think of it as being "small".

2. I assume that and and are upper and lower Riemann sums on some interval corresponding to some partition . In that case the point is that as the partition is refined the upper and lower sums approach one another and the limit is the Riemann integral.

3. From your questions it appears that you are focusing a bit too much on the symbol manipulations and not strongly enough on what they mean -- this is quite common in one's first exposure to rigorous mathematical analysis.

The idea behind limits is that the difference in the value of some function at two points can be made small (less than ) if the difference between and can be made sufficiently small (less than ).

The idea behind limits is that the difference in the value of some function at two points can be made small (less than ) if the difference between and can be made sufficiently small (less than ).

Ah yes thank you, but some problems in trying to get to the nearest possible surely puzzles me alot, in his book.

3. From your questions it appears that you are focusing a bit too much on the symbol manipulations and not strongly enough on what they mean -- this is quite common in one's first exposure to rigorous mathematical analysis.

ill have to say High school textbooks seemed like play time mathematics, where proofs of formula's and theorems were never too clearly discussed.
Seriously its nothing like what we have learned in high school , the book like has proofs for like everything. Something very novel indeed. But i guess this is good practice for me now, before entering IB.

Originally Posted by Heinsbergrelatz

Okay, its concerened with michael spivak's book.

there is this one inequality or rather symbol i see, ever since the definition. what exactly does the denote here?? i think it stands for some small distance error, or error in distance towards the y-axis, it does not entirely define what the \epsilon stands for.

also this inequality;

,

i know what the inequality represents but, just dont get the proof provided by the book. The other proofs are alright, though still very challenging indeed.

can anyone explain this concern i have?

thank you

The problem i have is none of those sympols mean anything to me.
I hate 'maths' like that'

The epsilon is, I think, called the "epsilon environment" (just a word-to-word translation into English ).

I think it is used in statistics, but also in "nearing procedures" (how is it called when you won't get a definite number, but instead you have to come closer and closer to the result?).