2.5 Subsequences and cluster points  (Page 3/4)

Now, using the lemma, we can give the proof of the Bolzano-Weierstrass Theorem.

If $\left\{a_n\right\}$ is a sequence of real numbers, this theorem is an immediate consequence of part (4) of the preceding lemma.

If $a_n=b_n+c_{n}i$ is a sequence of complex numbers, and if $\left\{a_n\right\}$ is bounded, then $\left\{b_n\right\}$ and $\left\{c_n\right\}$ are both bounded sequences of real numbers.See [link] . So, by the preceding paragraph, there exists a subsequence $\left\{b_{n_k}\right\}$ of $\left\{b_n\right\}$ that converges to a real number $b.$ Now, the subsequence $\left\{c_{n_k}\right\}$ is itself a bounded sequence of real numbers, so there is a subsequence $\left\{c_{n_{k_j}}\right\}$ that converges toa real number $c.$ By part (2) of [link] , we also have that the subsequence $\left\{b_{n_{k_j}}\right\}$ converges to $b.$ So the subsequence $\left\{a_{n_{k_j}}\right\}=\{b_{n_{k_j}}+c_{n_{k_j}}i\}$ of $\left\{a_n\right\}$ converges to the complex number $b+ci;$ i.e., $\left\{a_n\right\}$ has a cluster point. This completes the proof.

There is an important result that is analogous to the Lemma above, and its proof is easily adapted from the proof of that lemma.

The Bolzano-Wierstrass Theorem shows that the cluster set of a bounded sequence $\left\{a_n\right\}$ is nonempty. It is also a bounded set itself.

The following definition is only for sequences of real numbers. However, like the Bolzano-Weierstrass Theorem, it is of very basicimportance and will be used several times in the sequel.

Let $\left\{a_n\right\}$ be a sequence of real numbers and let $S$ denote its cluster set.

If $S$ is nonempty and bounded above, we define $lim\; sup{a}_n$ to be the supremum $supS$ of $S.$

If $S$ is nonempty and bounded below, we define $lim\; inf{a}_n$ to be the infimum $infS$ of $S.$

If the sequence $\left\{a_n\right\}$ of real numbers is not bounded above, we define $lim\; sup{a}_n$ to be $\infty ,$ and if $\left\{a_n\right\}$ is not bounded below, we define $lim\; inf{a}_n$ to be $-\infty .$

If $\left\{a_n\right\}$ diverges to $\infty ,$ then we define $lim\; sup{a}_n$ and $lim\; inf{a}_n$ both to be $\infty .$ And, if $\left\{a_n\right\}$ diverges to $-\infty ,$ we define $lim\; sup{a}_n$ and $lim\; inf{a}_n$ both to be $-\infty .$

We call $lim\; sup{a}_n$ the limit superior of the sequence $\left\{a_n\right\}$ , and $lim\; inf{a}_n$ the limit inferior of $\left\{a_n\right\}.$

The notions of limsup and liminf are perhaps mysterious, and they are in fact difficult to grasp.The previous exercise describes them as the resultof a kind of two-level process, and there are occasions when this description is a great help. However, thelimsup and liminf can also be characterized in other ways that are more reminiscent of the definition of a limit. These other ways are indicated in the next exercise.