Less than infinity or Less or Equal to infinity

What is the difference between Less than infinity or Less or Equal to infinity?

We cannot substitute infinity anyway, and have to use limits. So does it make sense to write less and equal to infinity?

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Please provide some context. What sort of infinity are you talking about? What sort of quantities are you comparing to infinity? – Chris Eagle Jul 13 '12 at 12:42
@Chris Eagle, consider the most simple case when 0 < x < infinity. – superM Jul 13 '12 at 12:43
That's insufficient context. In what book or paper did you see the phrase "less or equal to infinity"? – J. M. Jul 13 '12 at 12:47
@J. M., this is the 'closest' example: math.stackexchange.com/questions/170308/…. Does it make any sence here or elsewhere? – superM Jul 13 '12 at 12:49
Beware your assumptions. There is such a beast as the extended real line. – Willie Wong Jul 13 '12 at 13:29

In the context of the real numbers $\infty$ denotes "larger than any $x\in\mathbb R$". It is often useful to have this as a possible index, to indicate this what happens "after".

For example in the context of measure theory, a positive measure is a function from sets into $[0,\infty]$. That is, some sets have infinite measure. In another related context we have $p$-norms for $p\in[1,\infty]$ which is also a well-defined notion.

When we then say that $p\leq\infty$ we mean that the following proposition would hold for any value of $p$, either finite or infinite. For example, $\|f+g\|_p\leq\|f\|_p+\|g\|_p$ holds for finite $p$ and for $p=\infty$.

On the other hand sometimes we would like to indicate something holds only for the finite values of $p$, e.g. if $1\leq p<\infty$ then $\ell_p$ is separable. It is not true anymore for $\ell_\infty$, despite the space itself is a well-defined object whose index is $\infty$.

Or for example if $A\subseteq\mathbb R$ and the $0<m(A)\leq\infty$ then there is a subset of $A$ which is not Lebesgue measurable. On the other hand, if $m(A)<\infty$ then there is some $n\in\mathbb N$ such that $m(A\cap[-n,n])=m(A)$.

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In the usual use, such as in calculus, infinity isn't a real number, so a real number can't be equal to infinity, though some things may equal to a number or to infinity, such as limits of functions, suprema of sets, etc. When used in such context, writing "the limit of the series < infinity", means that the series converge (to a real number). Writing "the supremum of the set <= infinity", means that the supremum might be a real number and might aswell not exist (meaning the the set isn't bounded).

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