Sequence convergence?

Question

Does anyone have anythoughts on how to prove or disprove this? Tryin to get some practice in before school starts.

Let {ai } be a sequence of real numbers. If this sequence converges, then it converges
to a unique limit.

Thanks

Answer 1

When writing a proof, always, ALWAYS, always start from precise definitions.



"A sequence of numbers has the limit L if, for any prescribed accuracy, there is a position in the sequence such that all terms after this position approximate L within the prescribed accuracy." (ref. below)



Most "proofs" that you will see use a Greek "e" (epsilon) as the small prescribe value for the prescribed value; the position is called N; numbers that come after the Nth number could be called "a_(N+h)" or "a_m", where m is any number greater than N, or something like that. The key word in the definition is "any".



It could look like:



There exists a convergence limit if (and only if)

For any (non-zero) epsilon, there exists a N such that for all m greater than N, the difference between a_m and L will be less than epsilon.



For the proof, one could proceed by contradiction (try to prove the opposite and end up with nonsense):

Let us suppose that a convergent sequence has two distinct limits L and K.



If L and K are distinct, then their difference is not zero. Let us call the absolute value of this difference "delta".



The convergence test (above) requires that the approximation can be found for "any" epsilon. Then, let us make epsilon = delta divided by 10.



We have made our sequence convergent, therefore there must exist a N, such that for m greater than N, a_m will be within epsilon of L and within epsilon of K. At most, a_n will be exactly half-way between L and K, at a distance of epsilon from each. This means that the distance between L and K can be, at most 2 epsilon.

Yet, we know that this distance is "delta" = 10 epsilon.



By supposing that there could exist 2 distinct limits, we have arrived at a contradiction. Therefore, there cannot be 2 distinct limits.

Answer 2

The answers so far are fine as far as they go.



Where you may want to go with this is to be able to

construct an actual proof of this fact, no matter how

logical it may/may not appear.



This is a perfect situation for Proof by Contradiction.

It's to be proven that in the described circumstance, the

sequence converges to a unique limit. ASSUME NOT.

Then there are 2 different real numbers A and B, each of

which is the limit of the sequence. Let L = | A - B | be the

non-zero distance between A and B. Let d = L/2. Then

from the definition of limit, the sequence is eventually

(meaning from some point on) inside the open interval

(A-d , A+d) AND inside the open interval (B-d , B+d) .

Because these open intervals are disjoint (don't overlap)

this cannot happen. Therefore the original assumption

that we have different limits must be FALSE. Therefore

there is only one limit. End of Proof.



A mathematician is expected to be able to put together

such straightforward proofs like this in his/her sleep, with

one hand tied behind his/her back. There are scads of

basic elementary facts about numbers and about sets to

give you practice with this. It is part of the math language.



Mathematicians love this and expect this, but it has a

very practical side. This teaches you to be able to think

something through and say Exactly the reason and the

full reason why something is true. If you wind up being

a mathematician in Industry you will need to be able to

explain stuff to non-mathematicians. If you can give the

reasons well enough to satisfy a fellow-mathematician,

you should be able to satisfy your boss as well. If you

wind up being a mathematician in a teaching setting,

you will want to be able to explain to your students why,

for instance, the Quadratic Formula works. If you are

a mathematician who chooses to branch out into, for

instance, bridge design, you want to be able to think

through a design task thoroughly and deeply and logically

enough so that the bridge will not fall down 40 years later.



Basically, we have to have people in the gene-pool who

can be counted on to think things through clearly and

completely, and present facts clearly and logically. That

is why mathematicians walk the earth.

Answer 3

if An ? A?0 as n ? ?, the sequence is divergent the fundamental attempt says that the area between x-axis and the curve, from a million to infinity has a tendency to a million because of the fact the sequence a million/n^2 is asymptotical such as the area, the sequence converges (not at a million, considering the fact that sequence and fundamental are diverse) while the sequence is convergent it incredibly is the sequence of the partial sums that converges to a fee that's the sum of the sequence in different phrases we upload phrases and notice what ensue to the sequence of partial sums The undesirable new is that the cut back of partial sums is in many cases impossible to locate it incredibly is why there are a number of tests, which help to determinate convergence or divergence in accordance to the sequence you're examining besides do not subject: I keep in mind that I had many issues as quickly as I studied sequence for the 1st time, the answer is doing many many workouts

Answer 4

The whole point of convergence is that it converges to a unique limit.



Think of the formal process for proving convergence: given epsilon, there exists a delta such that ... etc., etc.



The whole point of this formal process is that the sequence becomes ARBITRARILY CLOSE to a GIVEN, EXPLICIT LIMIT. That being so, it CAN'T get arbitrarily close to some FINITELY DIFFERENT limit. This is not only common sense, but also hard mathematical logic!



Live long and prosper.

Answer 5

I believe by definition of convergence, a series that converges will converge to some limit.. lets call it L. This L has to be unique because if a series converges it cant converge to two different values. (Otherwise it doesnt converge)



I was never strong with proofs, but I think this logic works.

Answer 6

Let lim ai = L and let lim ai = M

0 = lim (ai-ai) = lim ai - lim ai = L - M

so L=M