Real Analysis – Sequences

For the more mathematical inclined the process of assuming the existence of the real numbers and defining properties they have to respect may not be the most satisfactory way to go about it, but that’s just what we need in here. Remember that this is a blog on Physics and not on Mathematics.

After having introduced the set of the real numbers, ${ \mathbb{R} }$ and proving some statements about them it is time for us to move on in the study of real analysis.

As was previously stated we’ll do this using sequences. We chose this road because some results are easier to prove using sequences and then natural extensions to functions can be achieved.

 Definition 10 A sequence, ${ u_n }$, is a mathematical function that acts on ${ \mathbb{N} }$, or a subset of it, and takes values on ${ \mathbb{R} }$. Symbolically $\displaystyle u_n:\mathbb{N}\rightarrow \mathbb{R} \ \ \ \ \ (10)$

As an example of a sequences we have ${ u_n=\displaystyle\frac{1}{n}}$ defined for all the natural numbers greater than ${ 0 }$.

Its graphical representation may be:

 Definition 11 We’ll say that ${ u_n}$ has as as limit the number ${ a\in\mathbb{R}}$, and denote it by ${ u_n \rightarrow a \in \mathbb{R}}$ or ${ \lim u_n = a \in \mathbb{R}}$, if, for each ${ \delta > 0}$ there exists one natural number ${ k}$, from which the distance between ${ u_n}$ and ${ a}$ is smaller than ${ \delta}$. $\displaystyle \forall \delta > 0 \;\exists k \in \mathbb{N}:\; n\geq k \Rightarrow |u_n - a| < \delta \ \ \ \ \ (11)$

Let us give a concrete example: For the graph we showed, we can see that ${ u_n }$ has smaller and smaller values, and that, from its definition, ${ u_n}$ is always positive. So, we can see that ${ u_n=\displaystyle\frac{1}{n} \rightarrow 0}$.

For all of this to be mathematically sound, we need to prove that for all ${ \delta > 0}$ we could indeed find a natural number ${ k}$ for which ${ n > k \Rightarrow |u_n-0| < \delta}$.

In order to do this it usually helps to see this condition as a game played between two people. One of them is constantly choosing values for ${ \delta}$ and the other is saying from which order ${k}$ the distance between ${ u_n}$ and ${ a}$ will be smaller than the given ${ \delta}$.

At a given point the order-giving player is tired to answer to all the different ${ \delta}$ the other player is choosing and decides to find a general expression for ${ k}$ as a function of ${ \delta}$. If such an expression can be found than the game is won and ${ u_n}$ really has as limit the number ${ a}$.

As a general rule we can also say that as ${ \delta}$ becomes smaller as the ${ k}$ from which the definition of limit is verified gets larger.

 Definition 12 The range of a sequence ${ u_n}$ is the set ${ \{u_n:n \geq p\}}$.
 Definition 13 Following definition 12 we’ll say that a sequence is bounded above if the set of its terms, the range, is bounded above. In an analogous way one can also define a bounded below sequence and a bounded sequence. One will say that a given sequence is unbounded when it isn’t bounded.

As an example of a bounded sequence (and by definition also a bounded below and bounded above sequence) we have ${ u_n=(-1)^n}$.

As an example of an unbounded sequence we have ${ u_n=n}$

Let us now suppose that we have a bounded sequence ${ u_n}$. That it is to say that there exist two real numbers ${ a}$ and ${ b}$ so that ${ a \leq u_n \leq b \quad \forall n}$.

Now ${ |u_n|=u_n}$ or ${ |u_n|=-u_n}$. Since ${ u_n \leq b}$ and ${ -u_n \leq -a}$ we can define ${ \alpha=\text{max}\{b,-a \}}$ and we are left with ${ u_n\leq \alpha}$ and ${ -u_n \leq \alpha}$. Or in an equivalent way ${ |u_n| \leq \alpha}$. Thus if ${ u_n}$ is bounded, there exists ${ \alpha > 0}$ such as ${ |u_n| \leq \alpha\quad \forall n}$. Reciprocally if ${ -\alpha\leq u_n \leq \alpha}$, ${ u_n}$ is a bounded sequence.

 Definition 14 We’ll say that a given sequence is convergent if it tends to a finite limit. We’ll say that the sequence is divergent otherwise.
 Theorem 13 If ${ u_n}$ is convergent then it is bounded. $\displaystyle \exists a \in \mathbb{R}: \lim u_n=a \Rightarrow \exists \alpha \in \mathbb{R}:\,|u_n| \leq \alpha \ \ \ \ \ (12)$ Proof: Omitted. $\Box$

Take care that the converse of this theorem needs not to be a true statement. We only know that convergent sequences have to be bounded ones, but we know nothing about the nature of a given sequence if it is a bounded one.

As an example we can cite ${ u_n = (-1)^n }$ which is a bounded sequence but isn’t a convergent one.

In mathematical lingo we say that the condition of a sequence being bounded isn’t sufficient for it to be convergent.

We will now introduce the notions of neighborhood of a given point. Loosely speaking the neighborhood of a point, ${ a }$ is the set of points that are near him.

— 5.1. Neighborhoods —

 Definition 15 Given ${ a \in \mathbb{R}}$ and ${ \delta > 0}$, the neighborhood of ${ a}$ of radius ${ \delta}$ is the set of points in ${ \rbrack a- \delta, a+ \delta \lbrack}$ and is denoted by ${ V(a, \delta)}$.

As an example let’s apply the notion of neighborhood in the definition of limit:

${ \begin{array}{rcl} |u_n-a| < \delta &\Leftrightarrow& a-\delta < u_n < a + \delta \\ &\Leftrightarrow& u_n \in \rbrack a-\delta,a+ \delta \lbrack \\ &\Leftrightarrow& u_n \in V(a,\delta) \end{array} }$

Thus ${ \lim u_n=a}$ if and only if ${ \forall \delta > 0\, \exists k \in \mathbb{N}:\, n\geq k \Rightarrow u_n \in V(a,\delta)}$ where we used the definition of neighborhood and the previous calculation.

5 Responses to “Real Analysis – Sequences”

1. typo-spotting:

– in the first paragraph after the last graph:
$$|u_n| \leq 0\quad \forall n$$
should be:
$$|u_n| \leq \alpha \quad \forall n$$

– In theorem I, it should be then and not “than”.

2. When do we get to the physics? 😛

3. My good man due to popular demand I’ll start the chapter on Newtonian Physics earlier than expected. Just as note I’ll expect people to know a little bit on differential and integral calculus. :p

And thanks a lot on the typos. Just keep’em coming as you find them. 😉

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5. […] April 3, 2009 After introducing sequences and gaining some knowledge of some of their properties (I,II ,III , and IV) we are ready to embark on the study of real analysis while using concepts that […]