Intuition: If $a\leq b+\epsilon$ for all $\epsilon>0$ then $a\leq b$? I am reading Tom Apostol's Analysis and come across this theorem. 

Should $a \leq b$ if $a\leq b+\epsilon$ for all $\epsilon &gt;0$? 

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.$\ \ $
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 $\varepsilon &gt;0$. 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 choices left are the numbers to the left or $b$ itself.
Assume for contradiction $a &gt; b$. However we can pick $\epsilon = (a-b)/2 &gt; 0$ so that $a &gt; b + \epsilon$, a contradiction of our original assumption $a \le b + \epsilon$. Imagine the real line:

$$
....\underbrace{b ......... b+\epsilon}_{\text{dist of } (a-b)/2} ......... a ....
$$

The same trick works for $a - \epsilon \le b + \epsilon$ using $\epsilon = (a-b)/3$.