About the using of $\le$ in an inequality

Question

Suppose that $x_1\le 4\lt x_2$.

Observe that $x_1$ will never be equal to $x_2$,

Can I still use $x_1\le x_2$ in any proof, is this always correct?

Answer 1

Yes, you can use that notation. The given inequality strictly implies $\boxed{x_1<x_2}$. Now if you write $x_1 \le x_2$, then it indicates $\boxed{x_1<x_2 \lor x_1=x_2}$. And realise that, for a "OR" statement, truth of either proposition mentioned in the statement implies truth of the statement. So what you want to write is totally true and correct.

Answer 2

If I have one urn with no more than $4$ beans in it, and I have another urn with more than four beans in it, then is it not true that the first urn has no more beans than the second one?