Sequences

In this section we revise the notion of convergence for real numbers and prove the Bolzano-Weierstrass property.

From section 1 we have a least upper bound (greatest lower bound) for any bounded increasing (respectively decreasing) sequence of real numbers.

Exercise 28   Show that any bounded non-decreasing sequence of real numbers has a least upper bound; a bounded non-increasing sequence has a greatest lower bound.

Now if S is any bounded non-empty set of real numbers, let c₁ be an upper bound and s₁ $\in$ S. Now we iteratively define a pair of sequences {s_n} and {c_n} as follows. If (s_n + c_n)/2 is an upper bound for S then we define s_{n + 1} = s_n and c_{n + 1} = (s_n + c_n)/2; otherwise let s_{n + 1} be any element of S so that s_{n + 1} > (s_n + c_n)/2 and c_{n + 1} = c_n.

Exercise 29   Show that {s_n} is a non-decreasing sequence of elements of S and {c_n} is a non-increasing sequence of upper bounds for S so that (c_{n + 1} - s_{n + 1}) $\leq$ (s_n - c_n)/2. Hence show that the greatest lower bound of {c_n} is equal to the least upper bound of {s_n} and this bound is the least upper bound for S.

In particular, if {x_n} is any bounded sequence of real numbers we have a least upper bound and greatest lower bound for this sequence. Let us define

$$\geq$$ l_k =

$$\geq$$ u_k =

liminf{x_n} = the least upper bound of {l_k}

limsup{x_n} = the greatest lower bound of {u_k}

Note that {l_k} is a non-decreasing sequence and {u_k} is a non-increasing sequence.

Exercise 30   Show that liminf{x_n} $\leq$ limsup{x_n}.

We say that the sequence {x_n} has a limit (is convergent) if these two numbers are equal; this number c is called the limit of this sequence of numbers.

Exercise 31   Show that for every positive $\epsilon$ there is a index n₀ so that | x_n - c| < $\epsilon$ for all n > n₀. Hence, there is an index n₁ so that | x_n - x_m| < $\epsilon$ for all n > n₁; this called Cauchy's criterion. Conversely show that any sequence {x_n} that satisfies Cauchy's criterion is convergent.

Now for any sequence {x_n} we can find subsequences {y_k = x_nk} (with n₁ < n₂ < ^... ) so that {y_k} is convergent (this is called the Bolzano-Weierstrass property).

Exercise 32   Show that there are subsequences of {x_n} that converge to liminf{x_n} and limsup{x_n}.

Finally, let us note some algebraic properties of convergent sequences.

Exercise 33   Show that the sum, difference and product of convergent sequences is limit of the sum, difference and product of the terms. If a sequence has a non-zero limit then show that the inverse of the limit is the limit of the inverses of the non-zero terms of the sequence.