19 - Bounded and unbounded sequences

Bounded (and Unbounded) Sequence

Let $(a_n)$ be a sequence

$(a_n)$ is bounded if the set $\{a_n:n\ge 1\}$ is bounded
In other words

$$\exists M\text{ such that}|a_n|\le M\text{ for all }n\ge 1$$

Otherwise it is unbounded

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Convergent Sequences are bounded

Let $(a_n)$ be a convergent sequence then

$$(a_n)\text{ is bounded}$$

Proof

Assume that $a_n\to L$ as $n\to \infty$
Then taking $ϵ=1$, there is $N$ such that if $n\ge N$ then $|a_n-L|<1$

$$|a_n|=|(a_n-L)+L|\le |a_n-L|+|L|<1+|L|$$

Let $M=\max \{|a_1|,|a_2|,\cdots ,|a_N|,|L|+1\}$
Then

$$|a_n|\le M\text{ for all }n\ge 1$$

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Sequence tending to infinity

Let $(a_n)$ be a real sequence
Then $(a_n)$ tends to infinity as $n\to \infty$ if

$$\forall M\in \mathbb{R}\exists N\in \mathbb{N}\text{ such that }\forall n\ge N,a_n>M$$

Also written as $a_n\to \infty$ as $n\to \infty$

Negative infinity $(a_n)$ tends to negative infinity as $n\to \infty$ if

$$\forall M\in \mathbb{R}\exists N\in \mathbb{N}\text{ such that }\forall n\ge N,a_n<M$$

Also written as $a_n\to -\infty$ as $n\to \infty$

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Limits of $n^{\alpha }$ lemma

1. If $\alpha <0$, then $n^{\alpha }\to 0$ as $n\to \infty$

2. If $\alpha >0$, then $n^{\alpha }\to \infty$ as $n\to \infty$

Proof

1. Take $ϵ\in (0,1)$ we have

$$\begin{array}{rl}{n}^{\alpha }& <ϵ\\ ⟺e^{\alpha \log n}& <ϵ\\ ⟺\alpha \log n& <\log ϵ\\ ⟺\log n& >\frac{1}{\alpha }\log ϵ(\text{note }\alpha <0)\\ ⟺n& >e^{\frac{1}{\alpha }\log ϵ}\end{array}$$

So let $N=1+\lceil e^{\frac{1}{\alpha }\log ϵ}\rceil$
2) Take $M>0$ we have

$$\begin{array}{rl}{n}^{\alpha }& >M\\ ⟺e^{\alpha \log n}& >M\\ ⟺\alpha \log n& >\log M\\ ⟺\log n& >\frac{1}{\alpha }\log M(\text{note }\alpha >0)\\ ⟺n& >e^{\frac{1}{a}\log ϵ}\end{array}$$

So let $N=1+\lceil e^{\frac{1}{a}\log ϵ}\rceil$

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Limits of $c^n$ ($c\in {\mathbb{R}}^{>0}$)

1. If $c<1$, then $c^n\to 0$ as $n\to \infty$

2. If $c=1$, then $c^n\to 1$ as $n\to \infty$

3. If $c>1$, then $c^n\to \infty$ as $n\to \infty$

Proof

1. Shown in Useful Convergences (1)
2. Clear from definition as $c^n=1$ for all $n\in \mathbb{N}$

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