- #1
Magenta
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I cannot remember if infinity is an upper bound for a subset of R?
I think so, but I want to be sure before I use it in a proof.
I think so, but I want to be sure before I use it in a proof.
In summary, a set is said to be bounded by infinity when its elements have no specific upper limit or maximum value. It cannot be both bounded and unbounded by infinity, and we can determine if a set is bounded by infinity by looking at its elements. Not all infinite sets are bounded by infinity, and this concept is closely related to the continuous and unbounded nature of real numbers.
- #2
tiny-tim
Science Advisor
Homework Helper
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Welcome to PF!
Hi Magenta ! Welcome to PF!
It depends what you mean by R.
Infinity isn't usually a member of R, and if it isn't then it can't be an upper bound.
Magenta said:
Hi Magenta ! Welcome to PF!
It depends what you mean by R.
Infinity isn't usually a member of R, and if it isn't then it can't be an upper bound.
Related to Can sets be bounded by infinity?
1. What does it mean for a set to be bounded by infinity?
When we say that a set is bounded by infinity, we mean that the elements in the set do not have a specific upper limit or maximum value. In other words, the set has an infinite number of elements that are continuously increasing without ever reaching a limit.
2. Can a set be both bounded and unbounded by infinity?
No, a set cannot be both bounded and unbounded by infinity. A set is either bounded, meaning it has a specific upper and lower limit, or unbounded, meaning it has no specific limits. If a set is unbounded, it cannot be bounded by infinity because there is no limit to the elements in the set.
3. How do we determine if a set is bounded by infinity?
To determine if a set is bounded by infinity, we can look at the elements in the set and see if they have a specific upper limit or maximum value. If the elements continue to increase without ever reaching a limit, then the set is bounded by infinity.
4. Are all infinite sets bounded by infinity?
No, not all infinite sets are bounded by infinity. Some infinite sets have specific limits, such as the set of all even numbers. This set has an infinite number of elements, but they are all bounded by the limit of 2.
5. How does the concept of boundedness by infinity relate to real numbers?
The concept of boundedness by infinity is closely related to the concept of real numbers. Real numbers are continuous and have no gaps, meaning they can be infinitely divided without ever reaching an end. This is similar to how a set bounded by infinity has an infinite number of elements that continue to increase without ever reaching a limit.
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