Supersequence (terminology)

**Swlabr** · April 1st 2010, 02:01 AM

If $A \subset B$ with $A, B$ sets then $A$ is a subset, $B$ a superset. My question is, do we use the same terminology for sequences? If $\{x_j\}_{j \in J}$ is a subsequence of $\{x_i\}_{i \in I}$ does that mean that $\{x_i\}_{i \in I}$ is a supersequence?

If not, what do we call it? Surely it must have a name!

Wikipedia thinks supersequence is a geological term, while the Penguin Dictionary of Maths omits both superset and supersequence...

**Tinyboss** · April 1st 2010, 06:05 AM

In my experience, "superset" is almost never used, though I have come across it before (usually not in a formal context, though). Instead of saying "A is a superset of B", it's customary to just say, "B is a subset of A".

I've never heard "supersequence".

I suspect that, in both cases, it's because it's not particularly meaningful to talk about all the supersets or supersequences of a set or sequence. On the other hand, we commonly (though not necessarily) say "x<y" or "x>y", rather than "x<y" or "y<x" (i.e. we use both "less than" and "greater than", not just "less than"), because it is meaningful to talk about all the numbers greater than y.

**Drexel28** · April 1st 2010, 08:52 AM

Originally Posted by Swlabr

If $A \subset B$ with $A, B$ sets then $A$ is a subset, $B$ a superset. My question is, do we use the same terminology for sequences? If $\{x_j\}_{j \in J}$ is a subsequence of $\{x_i\}_{i \in I}$ does that mean that $\{x_i\}_{i \in I}$ is a supersequence?

If not, what do we call it? Surely it must have a name!

Wikipedia thinks supersequence is a geological term, while the Penguin Dictionary of Maths omits both superset and supersequence...

I've never heard of that notation. Easy fix? Just say prior to you argument "Define a supersequence of a sequence to be..."

Originally Posted by Tinyboss

In my experience, "superset" is almost never used, though I have come across it before (usually not in a formal context, though). Instead of saying "A is a superset of B", it's customary to just say, "B is a subset of A".

I've never heard "supersequence".

I suspect that, in both cases, it's because it's not particularly meaningful to talk about all the supersets or supersequences of a set or sequence. On the other hand, we commonly (though not necessarily) say "x<y" or "x>y", rather than "x<y" or "y<x" (i.e. we use both "less than" and "greater than", not just "less than"), because it is meaningful to talk about all the numbers greater than y.

Really? I don't think it's that rare. I use it when it's more presentable. It is analogous as the following:

While equivalent saying $\frac{1}{n+1}\geqslant\frac{1}{2n}\geqslant \frac{1}{2}\cdot 0=0$ is much more logically presentable then $0=\frac{1}{2}\cdot 0\leqslant \frac{1}{2n}\leqslant\frac{1}{n+1}$ when we are trying to prove $\frac{1}{n+1}\geqslant 0$ . Or at least it is for me. It's almost because the "subject" of the inequality is the $\frac{1}{n+1}$ and one usually begins with the "subject".

**Tinyboss** · April 1st 2010, 08:55 AM

I think maybe you misunderstood me. I meant that we do use both < and >, while we don't often talk about supersets and supersequences, because the former is more often meaningful than the latter.

**Swlabr** · April 1st 2010, 10:17 AM

Originally Posted by Tinyboss

I think maybe you misunderstood me. I meant that we do use both < and >, while we don't often talk about supersets and supersequences, because the former is more often meaningful than the latter.

You use subset and superset precisely like you would use former and latter. I really can't give a better, or simplier, reason than that! It is a word for referencing it. An example of usage:

"x is in the subset implies that x is in the superset".

What I want to say is something along the lines of

"...the subsequence converges and so the supersequence converges..."

I suppose I could just refer to it as "the sequence"...

**Tinyboss** · April 1st 2010, 10:20 AM

Originally Posted by Swlabr

YWhat I want to say is something along the lines of

"...the subsequence converges and so the supersequence converges..."

Watch out, that's not at all true!

**Swlabr** · April 1st 2010, 10:23 AM

Originally Posted by Tinyboss

Watch out, that's not at all true!

I had written "as it is Cauchy", but decided to leave that to the "..."...

**Drexel28** · April 1st 2010, 12:00 PM

Originally Posted by Swlabr

I had written "as it is Cauchy", but decided to leave that to the "..."...

If you are saying that every Cauchy sequence with a convergent subsequence is convergent then you're right. Is that what you're saying?

**Swlabr** · April 1st 2010, 12:37 PM

Originally Posted by Drexel28

If you are saying that every Cauchy sequence with a convergent subsequence is convergent then you're right. Is that what you're saying?

Well, that was an example of where I would use "supersequence", but it's not actually where I was trying to use it (I do know the Cauchy result holds).

Although now that I look at where I was trying to use it then it is actually pretty similar;

"Suppose $\{x_n + Y\}$ is a Cauchy sequence in $X/Y$ . Then if we show that some subsequence $\{x_m + Y\}$ converges to a limit in $X/Y$ we are done as the entire sequence will converge to the same limit as the supersequence is Cauchy."

Neat proofs with clumsy wordings are the way forward!

**Drexel28** · April 1st 2010, 12:48 PM

Originally Posted by Swlabr

Well, that was an example of where I would use "supersequence", but it's not actually where I was trying to use it (I do know the Cauchy result holds).

Although now that I look at where I was trying to use it then it is actually pretty similar;

"Suppose $\{x_n + Y\}$ is a Cauchy sequence in $X/Y$ . Then if we show that some subsequence $\{x_m + Y\}$ converges to a limit in $X/Y$ we are done as the entire sequence will converge to the same limit as the supersequence is Cauchy."

Neat proofs with clumsy wordings are the way forward!

What in the name of God does that problem mean? Is that a quotient space?

**Swlabr** · April 1st 2010, 01:02 PM

Originally Posted by Drexel28

What in the name of God does that problem mean? Is that a quotient space?

My problem isn't a maths-ey problem, it is a terminology one. Am I allowed to use the word "supersequence".

The context is what I posted above, which is halfway through a proof about quotient spaces and completeness.

**Drexel28** · April 1st 2010, 01:04 PM

Originally Posted by Swlabr

My problem isn't a maths-ey problem, it is a terminology one. Am I allowed to use the word "supersequence".

The context is what I posted above, which is halfway through a proof about quotient spaces and completeness.

Ok! Just didn't get the $X/Y$ reference, but you obviously now mean the quotient space concept with Banach spaces.

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