Convergence of a sequence
This page is to be phased out and the content moved to either more appropriate places or to Limit (sequence)
Like with continuity there are three forms for convergence of a Sequence
Given a sequence (an)nn=1
Contents
[hide]First form
Introductory form
∀ϵ>0∃N∈N:n>N⟹|an−a|<ϵ
Second form
Metric space form
∀ϵ>0∃N∈N:n>N⟹d(an−a)<ϵ
Third form
Topological form
∀Na∃N∈N:n>N⟹an∈Na
Cauchy Criterion
Convergence can be shown without knowing what exactly the sequence converges to, see the Cauchy criterion for convergence page
Note on norms
Recall from norm that we can simply define d∥⋅∥(x,y)=∥x−y∥
∀ϵ>0∃N∈N:n>N⟹∥an−a∥<ϵ
Is is worth noting because in Functional Analysis norms are considered and if we deal with a metric space we are inside a branch of topology
Interesting examples
fn(t)=tn→0 in ∥⋅∥L1
Using the ∥⋅∥L1
We see that ∥fn∥L1=∫10xndx=[1n+1xn+1]10=1n+1
This clearly →0