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I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...
I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.2 Existence Results ... ...
I need some help in understanding the proof of Theorem 5.12 ...Theorem 5.12 and its proof read as follows:
View attachment 9501In the above proof by Andrew Browder we read the following:
" ... ... [For instance, one can choose a positive integer $$n$$ such that $$n \gt [f(b) - f(a) + 1](b - a) / \epsilon$$ ... ... "My question is as follows:
Why does Browder have $$+1$$ in the expression $$[f(b) - f(a) + 1](b - a) / \epsilon$$ ... ... ?Surely $$[f(b) - f(a)](b - a) / \epsilon$$ will do fine ... since ...
$$\mu ( \pi ) = (b - a)/ n$$
and so
$$\mu ( \pi ) [f(b) - f(a)] = [f(b) - f(a)] (b - a)/ n \lt \epsilon$$ ...
... so we only need ...
$$n \gt [f(b) - f(a)](b - a) / \epsilon$$
Hope someone can help ...
Peter
I am currently reading Chapter 5: The Riemann Integral and am currently focused on Section 5.2 Existence Results ... ...
I need some help in understanding the proof of Theorem 5.12 ...Theorem 5.12 and its proof read as follows:
View attachment 9501In the above proof by Andrew Browder we read the following:
" ... ... [For instance, one can choose a positive integer $$n$$ such that $$n \gt [f(b) - f(a) + 1](b - a) / \epsilon$$ ... ... "My question is as follows:
Why does Browder have $$+1$$ in the expression $$[f(b) - f(a) + 1](b - a) / \epsilon$$ ... ... ?Surely $$[f(b) - f(a)](b - a) / \epsilon$$ will do fine ... since ...
$$\mu ( \pi ) = (b - a)/ n$$
and so
$$\mu ( \pi ) [f(b) - f(a)] = [f(b) - f(a)] (b - a)/ n \lt \epsilon$$ ...
... so we only need ...
$$n \gt [f(b) - f(a)](b - a) / \epsilon$$
Hope someone can help ...
Peter