Sequences And Convergence
Sequences and Convergence
Let x1 , x2 , ..., xn , ... denote an infinite sequence of elements of a metric space
(S, d). We use {xn }∞ n=1 (or simply {xn }) to denote such a sequence.
Definition 1 Consider x0 ∈ S. We say that the sequence {xn } converges to x0 when n tends to infinity iff: For all > 0, there exists N ∈ N such that for all n > N , d(xn , x0 ) <
We denote this convergence by lim xn = x0 or simply xn −→ x0 . n→∞ Example 2 Consider the sequence {xn } in R, defined by xn = n1 . Then xn −→
0.
The way to prove this is standard: fix > 0. We need to find N ∈ N such that for all n > N , d(xn , 0) < . We have d(xn , 0) = |xn − 0| = | n1 |
So it is enough that n1 < , or equivalently n > 1 . So choosing N > 1 we know that for all n > N , d(xn , 0) < .
The fact that we define the concept of convergence does not imply that every sequence converges. This is illustrated in the next two examples. Let’s begin with a remark about what it means for a sequence {xn } not to converge to x0 .
Remark: To know what the non-convergence of a sequence means, we need to write the negation of the definition of convergence. That reduces to: There exists > 0, such that for all N ∈ N, there exists n > N such that d(xn , x0 ) ≥ .
For the ones of you familiar with propositional logic, notice that convergence to x0 can be written as
(∀ > 0)(∃N ∈ N)(∀n > N )d(xn , x0 ) <
Its negation is then given by
(∃ > 0)(∀N ∈ N)(∃n > N )d(xn , x0 ) ≥
Example 3 Consider the sequence {xn } in R, defined by xn = n. Then xn does not converge to any x0 .
Consider any x0 in R and = 12 (Notice that when we prove non-convergence, it is enough to find one > 0, and we do not need to do the argument for all ).
Then, for any N , just consider any n > max N, x0 + 1. Then, since n > x0 + 1, d(xn , x0 ) = |xn − x0 | > 1 > , and the result follows.
Example 4 Consider the sequence {xn } in R, defined by xn = (−1)n . Then xn does not converge.
It is obvious that we only need to check that xn does not converge to 1
Let x1 , x2 , ..., xn , ... denote an infinite sequence of elements of a metric space
(S, d). We use {xn }∞ n=1 (or simply {xn }) to denote such a sequence.
Definition 1 Consider x0 ∈ S. We say that the sequence {xn } converges to x0 when n tends to infinity iff: For all > 0, there exists N ∈ N such that for all n > N , d(xn , x0 ) <
We denote this convergence by lim xn = x0 or simply xn −→ x0 . n→∞ Example 2 Consider the sequence {xn } in R, defined by xn = n1 . Then xn −→
0.
The way to prove this is standard: fix > 0. We need to find N ∈ N such that for all n > N , d(xn , 0) < . We have d(xn , 0) = |xn − 0| = | n1 |
So it is enough that n1 < , or equivalently n > 1 . So choosing N > 1 we know that for all n > N , d(xn , 0) < .
The fact that we define the concept of convergence does not imply that every sequence converges. This is illustrated in the next two examples. Let’s begin with a remark about what it means for a sequence {xn } not to converge to x0 .
Remark: To know what the non-convergence of a sequence means, we need to write the negation of the definition of convergence. That reduces to: There exists > 0, such that for all N ∈ N, there exists n > N such that d(xn , x0 ) ≥ .
For the ones of you familiar with propositional logic, notice that convergence to x0 can be written as
(∀ > 0)(∃N ∈ N)(∀n > N )d(xn , x0 ) <
Its negation is then given by
(∃ > 0)(∀N ∈ N)(∃n > N )d(xn , x0 ) ≥
Example 3 Consider the sequence {xn } in R, defined by xn = n. Then xn does not converge to any x0 .
Consider any x0 in R and = 12 (Notice that when we prove non-convergence, it is enough to find one > 0, and we do not need to do the argument for all ).
Then, for any N , just consider any n > max N, x0 + 1. Then, since n > x0 + 1, d(xn , x0 ) = |xn − x0 | > 1 > , and the result follows.
Example 4 Consider the sequence {xn } in R, defined by xn = (−1)n . Then xn does not converge.
It is obvious that we only need to check that xn does not converge to 1