Serlo: EN: Subsequence

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Introductory Example

Datei:Teilfolgen - Einführendes Beispiel.webm Sometimes its necessary to speak about the "subsequence" of a sequence. Much like a subset is a "part" of a set, a subsequence is some part of a sequence -- all elements of a subsequence are also elements of the original sequence. A subsequence is in end effect constructed by removing some chosen terms of the original sequence. Regardless of how many terms are stripped from the original sequence, the resulting subsequence still has an infinite number of terms. For example, let's take the sequence $a_n=(-1{)}^n$:

We are interested in a subsequence composed of every other term of the original sequence$a_n$. This subsequence arises from either removing all terms with an even index or removing all terms with an odd index. If, for example, we remove all terms with odd index, we get the following schematic:

This gives rise to a subsequence that is constant $1$.

Mathematical Description

Datei:Teilfolgen – Erklärung der mathematischen Schreibweise.webm How can subsequences be denoted? First let's look at the indices of the sequence terms that we want to keep in the subsequence:

Now we want to find a sequence ${\left(n_k\right)}_{k\in \mathbb{N}}$ that describes these indices. In the above example we consider even indices. So the sequence can be written as $n_k=2k$:

We substitute this sequence into ${\left(a_n\right)}_{n\in \mathbb{N}}$. From there we get the subsequence ${\left(a_{n_k}\right)}_{k\in \mathbb{N}}$:

First we will build the sequence ${\left(n_k\right)}_{k\in \mathbb{N}}$ of relevant indices of the subsequence. We will set this subsequence into the original sequence ${\left(a_n\right)}_{n\in \mathbb{N}}$ for $n$ so that we get the sequence ${\left(a_{n_k}\right)}_{k\in \mathbb{N}}$.

In our example we have $n_k=2k$. So we substitute $2k$ for $n$ in $a_n=(-1{)}^n$. Then we obtain the subsequence $a_{2k}=(-1{)}^{2k}=1$.

Definition

Datei:Definition von Teilfolgen.webm

This concept is important for analysis since it is used to characterize so-called "limit points." However, these will not be properly defined and discussed until the next chapter.

Convergence of Subsequences

For subsequences we have the following important theorem:

The above theorem directly implies:

Application: convergence of mixed sequences

In the chapter „Beispiele und Eigenschaften von Folgen“ we have seen how to compose to sequences $(b_n{)}_{n\in \mathbb{N}}$ and $(c_n{)}_{n\in \mathbb{N}}$ to a "mixed sequence" $(a_n{)}_{n\in \mathbb{N}}$ . This mixture is defined as Vorlage:Einrücken That means, the sequence $(a_n{)}_{n\in \mathbb{N}}$ is composed out of the two subsequences $(a_{2n-1}{)}_{n\in \mathbb{N}}=(b_n{)}_{n\in \mathbb{N}}$ and $(a_{2n}{)}_{n\in \mathbb{N}}=(c_n{)}_{n\in \mathbb{N}}$ .

We may now ask how the convergence of the mixture $(a_n{)}_{n\in \mathbb{N}}$ relates to the convergences of its two constituents $(b_n{)}_{n\in \mathbb{N}}$ and $(c_n{)}_{n\in \mathbb{N}}$ . In order for $(a_n{)}_{n\in \mathbb{N}}$ to converge, two conditions must be satisfied:

• First, both subsequences $(b_n{)}_{n\in \mathbb{N}}$ and $(c_n{)}_{n\in \mathbb{N}}$ have to converge, as we know that for convergent sequences, all subsequences converge.
• Second, the limits of $(b_n{)}_{n\in \mathbb{N}}$ and $(c_n{)}_{n\in \mathbb{N}}$ must be identical. This is because if $(a_n{)}_{n\in \mathbb{N}}$ converges, then all of its subsequences must tend to the same limit.

If one of these two conditions is not satisfied, the mixed sequence $(a_n{)}_{n\in \mathbb{N}}$ must diverge. But are the two conditions also sufficient for convergence of the mixed sequence? Indeed, they are! We will now proof this. The limit of the mixed sequence must the coincide with the two limits of the subsequences.

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