Table of Contents
Can a sequence have infinite Subsequences?
Yes the subsequence must be infinite. Any subsequence is itself a sequence, and a sequence is basically a function from the naturals to the reals. Usually, this is the definition of subsequence.
Is the limit of a sequence a limit point?
The limit of a sequence is a point such that every neighborhood around it contains infinitely many terms of the sequence. The limit point of a set is a point such that every neighborhood around it contains infinitely many points of the set.
What sequence has an infinite number of terms?
An arithmetic infinite sequence is an endless list of numbers in which the difference between consecutive terms is constant. An arithmetic sequence can start at any number, but the difference between consecutive terms, called the common difference, must always be the same.
Can a sequence have multiple limit points?
They can also be called subsequential limits of the sequence, because it can be proved (fairly easily) that x is a cluster point of a sequence ⟨xn:n∈N⟩ if and only if some subsequence ⟨xnk:k∈N⟩ of ⟨xn:n∈N⟩ converges to x.
How many limits can a sequence have?
A sequence an has at most one limit: an → L and an → L′ ⇒ L = L′. Proof.
How many convergent subsequences can a sequence have?
For example take a sequence with n−1 ones and zeroes afterward. A sequence like an=1/n has uncountably infinitely many convergent subsequences since every subsequence is convergent and all the terms are distinct.
Do finite sets have limit points?
Any finite set is composed of isolated points only. Since for any isolated point there exists a neighborhood that does not contain any other element of the set, a finite set cannot have any limit points.
What is finite sequence and infinite sequence?
A sequence is finite if it has a limited number of terms and infinite if it does not. Since the sequence has a last term, it is a finite sequence. Infinite sequence: {4,8,12,16,20,24,…} The first term of the sequence is 4 . The “…” at the end indicates that the sequence goes on forever; it does not have a last term.
Can Infinity be a limit point?
In the usual topology the real numbers are homeomorphic to an open interval, and adding the two “endpoints” at infinity gives us a topological space that is homeomorphic to a closed interval. In that sense the notion of a (real) limit at infinity can be treated in a consistent way as a “point” at infinity.