Math for Non-Geeks/Limit theorems

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Limit proofs using the $ϵ$-definition are quite laborious. In this chapter we will study some limit theorems that simplifies matters.

Limit theorems

The limit theorems are the following:

Monotonicity rule: limit estimates

We also have the following monotonicity rule, which can be used to estimate limits:

Example: Computing the limit of a sequence

Consider the following sequence

This sequence is convergent. A proof using the $ϵ$-definition would be rather complicated. Fortunately we can break the whole sequence apart into individual sequences where the convergence and limits are known. For example $\underset{n\to \mathrm{\infty }}{\lim }\sqrt[n]{23}=1$. Using the limit theorems we can compute the limit step by step:

In this manner we can show that $(x_n{)}_{n\in \mathbb{N}}$ is convergent and the limit is $4$. Unfortunately this derivation is flawed: We are applying the limit theorems before we showed that the individual sequences are convergent. That those sequences do indeed converges becomes clear only after we have already employed the limit theorems. Therefore the above is not a valid proof. Instead we could proceed as follows:

We start with the sequences of which we know that they converge. By applying the limit theorems in reverse order we can derive the convergence and limit of the original sequence $(x_n{)}_{n\in \mathbb{N}}$. The symbol $\wedge$ is the logical conjunction, which should be read as "and".

Writing the proof in the above way is time-consuming and no fun. Most of the time we prefer writing down the first version. We use the limit theorems even though we don't know if the sequences converge. We must argue afterwards, that it was okay to use the limit theorems in the first place. But this is the case, because at the end, everything converges. Since the last steps worked, we were allowed to do the steps before. So if we want to write the proof like in the first version, we need to make sure at the end to give a justification why applying the limit theorems was a valid thing to do.

Problems with divergent sequences

As stated many times, we cannot use the limit theorems if one of the partial sequences diverges. If we forget this, we can quickly get nonsensical results:

Proof of limit theorems

Inversion of the absolute value rule

If $(|a_n|{)}_{n\in \mathbb{N}}$ is a null sequence, then the inversion of the absolute value rule holds, as well. From $\underset{n\to \mathrm{\infty }}{\lim }|a_n|=0$ , we have $\underset{n\to \mathrm{\infty }}{\lim }{a}_n=0$:

This inversion only holds for null sequences. For a general sequence, it may not hold. An example is the divergent sequence $a_n=(-1{)}^n$. The absolute value sequence $\underset{n\to \mathrm{\infty }}{\lim }|a_n|=1$ converges. So $\underset{n\to \mathrm{\infty }}{\lim }|a_n|=1$ but $\underset{n\to \mathrm{\infty }}{\lim }{a}_n\ne 1$. The latter limit even does not exist.

The root rule

The following rule is in fact a generalization to the power rule above. It extends its validity from integer powers (like $a^m,m\in \mathbb{N}$) to powers of the form $a^{1/k}=\sqrt[k]{a}$. Combining both rules, we get a limit theorem for all positive rational powers (like $a^{m/k}$).

Remarks concerning the monotonicity rule

There is a special case: we consider a constant $b_n=c$ :

The above proposition implies:

Both cases can be connected: „$a_n\le b_n$“ and „$a_n\ge b_n$“:

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