# Is there a symbol for always less than (or just always?)

For e.g, the quotient of $\frac{1}{n}$, $q$, where $n \gt 1$, $q$ will always be less than $1$.

$$\frac qn\le n$$

etc.

I can't really write $\frac {q}{n} < n$, because whilst true, it doesn't help provide much context. I am trying to write up proofs so I want to use mostly mathematical symbology.

Thanks :)

• $a<b$ usually gets the job done.. Apr 14, 2016 at 17:45
• "Am trying to write up proofs so I want to use mostly mathematical symbology." No!!! If you want to learn to write up proofs instead you should be learning to express yourself clearly in English. Apr 14, 2016 at 17:46
• Yes but I want to say that a<b is always true, not just sometimes. Apr 14, 2016 at 17:46
• "q/n will always be less or equal to n" - what do you mean by this? How does your q depend on n here? Secondly, "sometimes" is not a notion often used in a formal context, even if modal logic may capture it. Apr 14, 2016 at 17:47
• Look guys, you all got good points but I wasn't trying to write up a groundbreaking proof here, just need some help on nomenclature. Apr 14, 2016 at 17:48

The context is provided by what you write next to the equation (that is, what you write in words).

If you want to say "For all integers $n>1$ and all real $q<1$, we have $\frac q n<1$", then it is expected of you to write that.

If you are intent on using "symbols" then the "for all" symbol $\forall$ (whose partner "there exists", $\exists$, also comes in handy) is what you seek: $$\forall n\in\mathbb{Z},q\in\mathbb{R},\,\left(n>1,\,q<1\implies \frac qn<1\right)$$ However, as I hope is evident, it is often easier just to use words.

• And "easier just to use words" goes for the reader as well as for the writer. Apr 14, 2016 at 17:56

A proof, if you want to be formal, usually starts with given information. For example,

Given: a shape has $3$ sides

Then continues with steps to reach a goal; perhaps proving the shape is a triangle. These step have reasons, and as long as the reasons are sound, it is assumed when you give these steps, you are saying that they are always true within your given conditions:

Given: a shape has $3$ sides. The shape is a triangle, because of the definition of triangle, "a shape with $3$ side."

Again, we're not saying every shape is a triangle, but that every shape that fits our given conditions is a triangle. For your example,

Given $\frac qn$ such that $q = \frac 1n$ and $n>1$,

$\frac qn<n$ because...

You are stating in that case not that $\frac qn<n$ is always true, but that within your given conditions it is always true. You simply need to back it up with reasons that are already accepted as true or previously proven.

Hope that helps!