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A monotone increasing sequence {r_n | n>0} is a sequence with: n>m implies r_n >= r_m
A monotone decreasing sequence {r_n | n>0} is a sequence with: n>m implies r_n <= r_m
A strictly monotone increasing sequence {r_n | n>0} is a sequence with: n>m implies r_n > r_m
A strictly monotone decreasing sequence {r_n | n>0} is a sequence with: n>m implies r_n < r_m
Theorem. All bounded monotone sequences of real numbers have a unique limit.
What else can I help you with?
Define or describe each set of real numbers?
define or describe each set of real numbers?
What are the component of real numbers?
Real Numbers cannot be the square root of a negative number. Real Numbers are not divided by zero. Basically, Real Numbers cannot be anything that is undefined.
What is hierarchy of real numbers?
a real numbers computable if it is limit of an effectively converging computable sequence of a retional supremum infimum computable if it is supremum of computable of sequence of a rational numbers
What is define real number?
It is a bit hard to define them - and the exact definitions are a bit formal. It is best to think of real numbers as the equivalent of all points on a straight line, infinite in both directions.
What is the definition of the word real number?
a real number is a rational number or the limit of a sequence of rational numbers, as opposed to a complex number.
Is 867-5309 a real number?
6176
How do you prove that if a real sequence is bounded and monotone it converges?
We prove that if an increasing sequence {an} is bounded above, then it is convergent and the limit is the sup {an }Now we use the least upper bound property of real numbers to say that sup {an } exists and we call it something, say S. We can say this because sup {an } is not empty and by our assumption is it bounded above so it has a LUB.Now for all natural numbers N we look at aN such that for all E, or epsilon greater than 0, we have aN > S-epsilon. This must be true, because if it were not the that number would be an upper bound which contradicts that S is the least upper bound.Now since {an} is increasing for all n greater than N we have |S-an|
Are most numbers rational or irrational?
The set of irrational numbers is larger than the set of rational numbers, as proved by Cantor: The set of rational numbers is "countable", meaning there is a one-to-one correspondence between the natural numbers and the rational numbers. You can put them in a sequence, in such a way that every rational number will eventually appear in the sequence. The set of irrational numbers is uncountable, this means that no such sequence is possible. All rational and irrationals (ie real numbers) are a subset of complex numbers. Complex numbers, in turn, are part of a larger group, and so on.
Why rational numbers are dense then real numbers?
Roughly speaking, rational numbers can form real numbers that's why they are more densed than real numbers. For example, if A is a subset of some set X & every point x of X belongs to A then A is densed in X. Also Cauchy sequence is the best example of it in which every bumber gets close to each other hence makes a real number.
What is the domain and range of y equals cubed square root of x?
The simplest answer is that the domain is all non-negative real numbers and the range is the same. However, it is possible to define the domain as all real numbers and the range as the complex numbers. Or both of them as the set of complex numbers. Or the domain as perfect squares and the range as non-negative perfect cubes. Or domain = {4, pi} and range = {8, pi3/2} Essentially, you can define the domain as you like and the definition of the range will follow or, conversely, define the range and the domain definition will follow,
What number has no multiple inverse?
On the set of all real numbers ZERO has no multiplicative inverse. For other sets there may be other numbers too, so please define your set!
364 361 19 16 4 1?
The next 2 numbers are 1 and -2 after which, in the real domain, the sequence stops.
