Serlo: EN: Unbounded sequences diverge

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In this chapter we will see that unbounded sequences must diverge. From this we can follow that convergent sequences must be bounded.

Unbounded sequences diverge

In the chapter „Konvergenz und Divergenz beweisen“ we have already seen that the sequence ${\left(2^n\right)}_{n\in \mathbb{N}}$ diverges. We used the property that the sequence grows beyond any boundary. That is to say if we have $a\in ℝ$ fixed, then there exists $N\in \mathbb{N}$ with $2^N\ge a+1$. Also for all $n\in \mathbb{N}$ with $n\ge N$ we have $2^n\ge a+1$ and thus

Infinitely many members ${\left(2^n\right)}_{n\in \mathbb{N}}$ lie outside of the $ϵ$-neighbourhood $(a-1;a+1)$. Therefore ${\left(2^n\right)}_{n\in \mathbb{N}}$ cannot converge towards $a$. If that were the case almost all members ${\left(2^n\right)}_{n\in \mathbb{N}}$ would have to be contained inside $(a-1;a+1)$, which is not the case. Because $a$ was chosen arbitrary, the sequence ${\left(2^n\right)}_{n\in \mathbb{N}}$ cannot have a limit and is therefore divergent.

We can extend this argument to any sequence that is unbounded, since we only used the property that ${\left(2^n\right)}_{n\in \mathbb{N}}$ becomes arbitrary large. Remember the definition of an unbounded sequence:

We can use this property to prove the following theorem:

Convergent sequences are bounded

Proof by contraposition

The above theorem tells us that unbounded sequences are divergent. With the aid of logical contraposition, we can follow that convergent sequences must be bounded. The principle of contraposition is:

The theorem is the following implication

Using contraposition we obtain the equivalent statement:

But this means that

If you are not sure that contraposition works just make the truth table of $(A⟹B)$ and compare it to that of $(\mathrm{\neg }B⟹\mathrm{\neg }A)$. As an example: "If it rains ($A$), the ground is wet ($B$)." The contraposition is: "If the gound is not wet ($\mathrm{\neg }B$), it doesn't rain ($\mathrm{\neg }A$)." This two statements are logically equivalent.

So by contraposition the following theorem is true, which we will need to prove further results later on:

Alternative direct proof

We also want to show an alternative, direct way of proving that a convergent sequences is bounded. This proof is often given in other textbooks. It shows how one can use the $ϵ$-definition of the limit in a proof.

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