Sequence question (analysis)

Hi guys,

I'm working on a complex analysis problem which states that I have

"a sequence {an} of distinct points in some region G such that an --> a in G as n --> infinity"

(then I'll need to do something with the sequences)

First question: Is **{a****n}** an infinite sequence? (In general, when I see this notation in analysis should I always assume an infinite sequence?)

I'm going to assume that it is an infinite sequence, then, I have a function represented by, lets say, **f(an) = cos(an)**

Since all sequences converge to the point a, is it true that **f(an) = f(a) = cos(a)**?

Do I have a constant function? That's what it looks like to me.

Thanks!

Re: Sequence question (analysis)

Quote:

First question: Is **{a****n}** an infinite sequence? (In general, when I see this notation in analysis should I always assume an infinite sequence?)

In this case, yes. In general, I do think of this as an infinite sequence. There may be exceptions, however, and depends on context.

Quote:

Since all sequences converge to the point a, is it true that **f(an) = f(a)**?

Not in general. Consider the function $\displaystyle f(x)=1$ if $\displaystyle x\not= a$, $\displaystyle f(x)=0$ otherwise. The infinite sequence $\displaystyle a_n=\frac{1}{n}$ has limit $\displaystyle a=0$ and satisfies $\displaystyle f(a_n)=1$ for all $\displaystyle n$. But $\displaystyle f(0)=0$. You need stronger conditions (i.e. function is continuous) for this statement to be true. For your example, it would be so since $\displaystyle \cos(z)$ is analytic.

Re: Sequence question (analysis)

Thanks Gusbob.

My function is, in fact, homomorphic on region G so I think that would make the previous statement true? Cos(zn) was just an example and it happened to be meet the criteria.

EDIT: Saw your edit. Thanks.

Re: Sequence question (analysis)

One more question with regards to the original, convergent sequence {zn} of distinct points.

Another problem asks whether there exists a function on region G such that **f(zn) = n for all n**.

I have trouble interpreting what they're asking. What is **n** in the output? If input to function is a sequence then I would expect the output to be a sequence. Could **n** be the limit of the output sequence?

(If the input to the function was a single element of the sequence, it would make more sense to me)

Thanks