Can a Non-Decreasing Function Have a Limit at Infinity?

[Legendre]

I read that "if f : R -> R is an increasing function, then limit as x tend to infinity of f(x) is either infinity, minus infinity or a real number". f an increasing function means { x < y } => { f(x) < or = f(y) }.

How do I prove this (if it is true)? Can I apply this to a function g : R -> [0,1]?

P.S.
I am not looking for a precise proof. A loose discussion of the way to prove this would be fine.

Thanks!

 

[HallsofIvy]

Actually, such a function cannot have a limit of negative infinity. If the function is bounded (there exist a number M such that f(x)< M for all x) then f(x) converges to a real number. If it is unbounded, then it goes to positive infinity.

How you would prove that depends on what you have available. Can you use the "least upper bound property"- that every nonempty set of real numbers with an upper bound has a least upper bound? If so then: Since f(x)< M for all x, the set {f(x)} has M as upper bound and so has a least upper bound, \alpha. You can use an "\epsilon, N" argument to show that \alpha is the limit.

If the function is not bounded, then, given any Y> 0, there exist x₀ such that f(x₀)> Y. But then if x₁> x₀, because f is increasing, f(x₁)> f(x₀)> Y.

 

[Matterwave]

What other limits can there possibly be, other than infinity, minus infinity, or a real number?

The function obviously is from the real plane to the real plane...so there can be no other limits.

Are you perhaps asking to prove that such a limit exists?

 

[elibj123]

There can be no limit at all.

For example sin(x) has no limit at infinity (proven simply with Heine Theorem)

As also xsin(x), although it seems it goes to infinity, it will actually oscillate between very large negative numbers and very large positive numbers (to prove this, Heine's Theorem won't work, and the resort is the original definition of the limit)

 

[Legendre]

Actually, such a function cannot have a limit of negative infinity. If the function is bounded (there exist a number M such that f(x)< M for all x) then f(x) converges to a real number. If it is unbounded, then it goes to positive infinity.

Great idea. However, I am terribly sorry if I didn't make it clear that what I meant by "increasing" is not "strictly increasing". Perhaps I should have mention that f is non-decreasing in the first place!

The case that I have trouble with is when f is NOT strictly increasing and yet IS non-decreasing.