Limit superior, limit inferior

-

Some sequences of practical importance (in science and engineering) may not have limits but have limits superior or limits inferior. While converging sequences are bounded, bounded sequences are not always converging. However, bounded sequences always have limits superior and limits inferior.




First, let us review some basic notions.

Definition (Upper bound, lower bound)

Let \(S\) be a subset of \(\mathbb{R}\): \(S \subset \mathbb{R}\). Let \(\alpha \in \mathbb{R}\) be a real number
  1. If we have \(x \leq \alpha\) for all \(x \in S\), then \(\alpha\) is said to be an upper bound of \(S\). 
  2. If we have \(x \geq \alpha\) for all \(x \in S\), then \(\alpha\) is said to be a lower bound of \(S\). 

Definition (Supremum, infimum)

Let \(S\) be a subset of \(\mathbb{R}\): \(S \subset \mathbb{R}\).
  1. If \(S\) is bounded above, and there exists the lowest value of the upper bounds, then the lowest upper bound is called the supremum of \(S\), denoted \(\sup S\). That is, if \(U(S)\) represents the set of all upper bounds of \(S\), then \[\sup S = \min U(S).\]
  2. If \(S\) is bounded below, and there exists the largest value of the lower bounds, then the largest lower bound is called the infimum of \(S\), denoted \(\inf S\). That is, if \(L(S)\) represents the set of all lower bounds of \(S\), then \[\inf S = \max L(S).\]

Let \(\{a_n\} = \{a_1, a_2, \cdots\}\) be a bounded sequence. For any \(n \in \mathbb{N}\), we define \(\overline{a}_n\) and \(\underline{a}_n\) by

\[\begin{eqnarray} \overline{a}_n &=& \sup_{k\geq n}a_k = \sup\{a_k \mid k \geq n\},\\ \underline{a}_n &=& \inf_{k\geq n}a_k = \inf\{a_k \mid k \geq n\}. \end{eqnarray}\]
That is, \(\overline{a}_n\) and \(\underline{a}_n\) are the supremum and infimum, respectively, of the subsequence \(a_n, a_{n+1}, a_{n+2},\cdots\). Clearly, we have
\[\underline{a}_n \leq a_n \leq \overline{a}_n\]
for all \(n\in\mathbb{N}\). Noting that
\[\{a_k \mid k \geq n\} \supset \{a_k \mid k \geq n+1\},\]
we have
\[\begin{eqnarray} \overline{a}_n &\geq & \overline{a}_{n+1},\\ \underline{a}_n &\leq & \underline{a}_{n+1}. \end{eqnarray}\]
Thus, \(\{\overline{a}_n\}\) and \(\{\underline{a}_n\}\) are monotone decreasing and increasing, respectively, and we have the following inequalities:
\[\underline{a}_1 \leq \underline{a}_2 \leq \cdots \leq  \underline{a}_n \leq \cdots \leq \overline{a}_n \leq \cdots \leq \overline{a}_2 \leq \overline{a}_1.\]
Therefore,
  1. The sequence \(\{\overline{a}_n\}\) is monotone decreasing and bounded.
  2. The sequence \(\{\underline{a}_n\}\) is monotone increasing and bounded.
Recall that any bounded monotone sequence converges. Thus, the following notions of limit superior and limit inferior are well-defined.

Definition (Limit superior and limit inferior)

Let \(\{a_n\}\) be a bounded sequence. We define the limit superior of \(\{a_n\}\) by
\[\limsup_{n\to\infty}a_n = \lim_{n\to\infty}\sup_{k\geq n}a_k,\]
and the limit inferior by
\[\liminf_{n\to\infty}a_n = \lim_{n\to\infty}\inf_{k\geq n}a_k.\]

Remark. Using \(\overline{a}_n = \sup_{k\geq n}a_n\) and \(\underline{a}_n = \inf_{k\geq n}a_n\), we can also write
\[\begin{eqnarray*} \limsup_{n\to\infty}a_n &=& \lim_{n\to\infty}{\overline{a}_n},\\ \liminf_{n\to\infty}a_n &=& \lim_{n\to\infty}{\underline{a}_n}. \end{eqnarray*}\]
□
Remark. Some authors prefer to denote the limit superior and limit inferior as
\[\varlimsup_{n\to\infty}a_n\]
and
\[\varliminf_{n\to\infty}a_n,\]
respectively. □

Example. Let's see a few examples.
  1. If \(a_n = (-1)^{n+1}\), then \(\limsup_{n\to\infty}a_n = 1\) and \(\liminf_{n\to\infty}a_n = -1\).
  2. If \(a_n = \frac{(-1)^n}{n}\), then \(\limsup_{n\to\infty}a_n = \liminf_{n\to\infty}a_n = 0\).
  3. If \(\{a_n\}\) is bounded and monotone increasing with \(\alpha = \sup_{n\in\mathbb{N}}a_n\), then \(\limsup_{n\to\infty}a_n = \liminf_{n\to\infty}a_n = \alpha\).
□

See also: Limit inferior and limit superior (Wikipedia).

Comments

Post a Comment

Popular posts from this blog

Open sets and closed sets in \(\mathbb{R}^n\)

-
Image
Open sets In \(\mathbb{R}\), we have the notion of an open interval such as \((a, b) = \{x \in \mathbb{R} | a < x < b\}\). We want to extend this idea to apply to \(\mathbb{R}^n\). We also introduce the notions of bounded sets and closed sets in \(\mathbb{R}^n\). Recall that the \(\varepsilon\)-neighbor of a point \(x\in\mathbb{R}^n\) is defined as \(N_{\varepsilon}(x) = \{y \in \mathbb{R}^n | d(x, y) < \varepsilon \}\) where \(d(x,y)\) is the distance between \(x\) and \(y\). Definition (Open set) A subset \(U\) of \(\mathbb{R}^n\) is said to be an  open set  if the following holds: \[\forall x \in U ~ \exists \delta > 0 ~ (N_{\delta}(x) \subset U).\tag{Eq:OpenSet}\] That is, for every point in an open set \(U\), we can always find an open ball centered at that point, that is included in \(U\). See the following figure. Perhaps, it is instructive to see what is  not  an open set. Negating (Eq:OpenSet), we have \[\exists x \in U ~ \forall \delta > 0 ~ (N_{\delta}(x) \not
Read more

Euclidean spaces

-
Image
We would like to study multivariate functions (i.e., functions of many variables), continuous multivariate functions in particular. To define continuity, we need a measure of "closeness" between points. One measure of closeness is the Euclidean distance. The set \(\mathbb{R}^n\) (with \(n \in \mathbb{N}\)) with the Euclidean distance function is called a Euclidean space. This is the space where our functions of interest live. The real line is a geometric representation of \(\mathbb{R}\), the set of all real numbers. That is, each \(a \in \mathbb{R}\) is represented as the point \(a\) on the real line. The coordinate plane , or the \(x\)-\(y\) plane , is a geometric representation of \(\mathbb{R}^2\), the set of all pairs of real numbers. Each pair of real numbers \((a, b)\) is visualized as the point \((a, b)\) in the plane. Remark . Recall that \(\mathbb{R}^2 = \mathbb{R}\times\mathbb{R} = \{(x, y) | x, y \in \mathbb{R}\}\) is the Cartesian product of \(\mathbb{R}\) with i
Read more

Applications of multiple integrals

-
Image
We can use multiple integrals to compute areas and volumes of various shapes. Area of a planar region Definition (Area) Let \(D\) be a bounded closed region in \(\mathbb{R}^2\). \(D\) is said to have an area if the multiple integral of the constant function 1 over \(D\), \(\iint_Ddxdy\), exists. Its value is denoted by \(\mu(D)\): \[\mu(D) = \iint_Ddxdy.\] Example . Let us calculate the area of the disk \(D = \{(x,y)\mid x^2 + y^2 \leq a^2\}\). Using the polar coordinates, \(x = r\cos\theta, y = r\sin\theta\), \(dxdy = rdrd\theta\), and the ranges of \(r\) and \(\theta\) are \([0, a]\) and \([0, 2\pi]\), respectively. Thus, \[\begin{eqnarray*} \mu(D) &=& \iint_Ddxdy\\ &=&\int_0^a\left(\int_0^{2\pi}rd\theta\right)dr\\ &=&2\pi\int_0^a rdr\\ &=&2\pi\left[\frac{r^2}{2}\right]_0^a = \pi a^2. \end{eqnarray*}\] □ Volume of a solid figure Definition (Volume) Let \(V\) be a solid figure in the \((x,y,z)\) space \(\mathbb{R}^3\). \(V\) is sai
Read more