I am reading Tom Apostol's Analysis and come across this theorem.
Should $a \leq b$ if $a\leq b+\epsilon$ for all $\epsilon >0$?
I don't doubt the proof in the book but I don't understand the intuition or geometric explanation behind this. Could somebody shed some light on this equation? I just started studying analysis on my own.$\ \ $
Best Answer
Draw a number line. Mark the point $b$. Where can you mark $a$? Every number greater than $b$ may be written as $b+\varepsilon$ for some $\varepsilon >0$. Then $a\leqslant b+\varepsilon$ says every number greater than $b$ is also greater than $a$. Thus, you erase all what comes after $b$. The only remaining choices are the numbers to the left or $b$ itself.