A Natural Limit Definition

Often, the first exposure one gets to rigorous mathematics is the definition of a limit. Let’s consider what this is for a sequence. We say \lim_{n\rightarrow \infty} a_n = A   if

\displaystyle  \forall \epsilon \in \mathbb{R}^+\quad\exists N \text{  s.t.}\quad\quad n > N \implies |a_n-A| \leq \epsilon   

This, at first sight, is ugly. It takes a while to even understand what it’s saying, longer to see why it works, and much longer to apply it. It’s intimidating to say the least. I feel, however, there is another version that makes the idea of limits simple and very natural giving a deep insight into what a limit really is.

First, note we are considering the space l^\infty which is the set of all sequences a=\{a_n\} such that \sup_{n} |a_n|<\infty . Basically, it doesn’t blow up. For example, \{n\}\not\in l^\infty but \{(-1)^n\}\in l^\infty .

Now, consider the transformation \displaystyle a^{(m)} = \{a_{n+m}\} . For example, say a = \{4,7,3,8,1,2, \dots \} , then a^{(2)}=\{3,8,1,2,\dots\} .

Finally, we define a partial order (not all elements are comparable) on this sequence.

a < b \iff a_n < b_n \quad\forall n\in\mathbb{N}

With all this set up, we define the following.

Definition 1.1 The limit function L:l^\infty\rightarrow \mathbb{R} is a linear order-preserving* map such that

\displaystyle L[a] = L[a^{(m)}] \quad \forall m\in \mathbb{N}

\displaystyle L[1] = 1

* a \leq b \implies L[a]\leq L[b] .

Note that there is not much special about this function. Linearity is very common and order preserving is not that restrictive. The only special part is its invariance under shifting which, as it turns out, is enough to restrict it to exactly what we want.

Theorem 1.1 If the limit of a=\{a_n\} exists, \lim_{n\rightarrow \infty} a_n = L[a]

Proof  Let \lim_{n\rightarrow \infty} a_n = A . Then, by definition, \forall \epsilon\in\mathbb{R}^+, \exists N\in\mathbb{N} such that

\displaystyle n > N \implies |a_n-A| \leq \epsilon 

We use the sequence transformation to write this more concisely.

\displaystyle |a^{(N)}_n-A| \leq \epsilon

We now look at the limit function of the sequence |a^{(N)}_n-A| . By the inequality condition for the limit function, we know

\displaystyle L[|a^{(N)}-A|] \leq \epsilon

But of course, a^{(N)}_n-A < |a^{(N)}_n-A|   so

\displaystyle L[a^{(N)}-A] \leq \epsilon

Apply linearity and its transformation invariance.

\displaystyle L[a^{(N)}-A]=L[a^{(N)}]-A =L[a]-A .


\displaystyle L[a]-A \leq \epsilon

\epsilon can be any positive value so

\displaystyle L[a] - A\leq 0 \implies L[a]\leq A

Going back a few steps and noting that A-a^{(N)}_n \leq |a^{(N)}_n-A| as well, we get A \leq L[a] which concludes the proof.

\displaystyle \blacksquare

Amazing, isn’t it? So let’s back up for a second and think about what we have shown. The first two properties of linearity and order preservation basically translate to “it’s a nice function.” But there are SO many functions that could fit this description. Here, we have shown that invariance under arbitrarily deleting the first few elements is enough to restrict it down to our limit definition. This says something very deep: the limit can be defined as just any “nice” function that only cares what happens at the end of the sequence. And, in fact, this result makes the original definition believable because it satisfies properties that are very natural to our notion of what a limit is.

Even more interestingly, you can use this definition to uniquely define the limit of sequences that can’t be defined using the original limit definition. For example, sequences of the form a=\{S+D(-1)^n\}=\{S-D,S+D,S-D,S+D,S-D,\dots\}  . Consider the following.

\displaystyle 2S-a = \{S-D(-1)^n\} = \{S+D(-1)^{n+1}\} = a^{(1)}

From this, we calculate the limit.

\displaystyle L[2S-a]=2S-L[a]\stackrel{!}{=}L[a^{(1)}]=L[a]

\displaystyle \implies L[a] = S

This is a somewhat obvious result but people are usually hesitant to say that it makes any rigorous sense. Here, however, it was formed from something very rigorous and natural.

Let me take this moment to note that I am NOT suggesting that this should be any sort of replacement for the original definition. The original is profoundly better for proofs and is actually slightly different because it isn’t defined for these weird alternating sequences. This is vital to the notions of continuity in analysis and should not be tampered with.

Returning to it however, we can write a nice explicit formula for it.

\displaystyle L[a] = \lim_{n\rightarrow\infty} \frac{1}{n}\sum_{i=1}^n a_i

I leave it as an exercise to the reader to show it both always converges and follows the required properties.

Hopefully, this gave you a cool insight into how to think about limits. As a challenge, I invite you to try to extend this to other regimes like the limits for functions. Good luck!

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