If we could just use '<' instead of '≤', why are we still using '≤' in many statements?

Question

For example,in " $a\leq b+\epsilon$ if $\forall\epsilon>0$, then $a\leq b$ ",it's impossible to find a case where $a=b+\epsilon$ for very $\epsilon>0$.However,people are still using $\leq$ even though $<$ is enough to describe that. Why is that?

Answer 1

There may be many reasons, one could be that it is the more general statement. Another, that this way it is easier to remember, you don't need to remember which of the instances is $\le$ and which is $<$.

Answer 2

It is often a lot harder to show $<$ and a lot easier to show $\leq$. For example, if you have a sequence $(a_n)_{n \in \mathbb{N}}$ that is bounded above by a constant $C$, so $a_n \leq C$ for all $n$. If further the sequence converges, then also $$\lim\limits_{n \to \infty} a_n \leq C.$$

On the other hand, even if $a_n < C$ for all $n$, you can't conclude that $$\lim\limits_{n \to \infty} a_n < C.$$

Answer 3

The statement you ask about doesn't make sense as written, but presumably it is meant to say

If $a\leq b+\epsilon$ for every $\epsilon>0$ then $a\leq b$

In this case you can't replace the second $\leq$ by $<$ (even if you also change the first), since we could have $a=b$.

You could replace the first $\leq$ by $<$, but the original version is a stronger statement, so preferable. (For any given $\epsilon$, $a\leq b+\epsilon$ is a weaker condition than $a<b+\epsilon$, and making the conditions weaker gives a stronger statement.)