How does the bisection method construct sequences to approximate the supremum of a bounded set?

Q: How does the bisection method construct sequences to approximate the supremum of a bounded set?

The bisection method starts with an element $a_1$ in the set $M$ and an upper bound $b_1$. It iteratively calculates the midpoint $c_n = \frac{1}{2}(a_n + b_n)$. If $c_n$ is an upper bound for $M$, it becomes the new upper bound $b_{n+1}$; otherwise, an element $x \in M$ greater than $c_n$ is found and becomes the new lower bound $a_{n+1}$. This recursively generates two sequences $(a_n)$ and $(b_n)$ that narrow the interval containing the supremum.

The bisection method starts with an element

a_1

in the set

M

and an upper bound

b_1

. It iteratively calculates the midpoint

c_n = \frac{1}{2}(a_n + b_n)

. If

c_n

is an upper bound for

M

, it becomes the new upper bound

b_{n+1}

; otherwise, an element

x \in M

greater than

c_n

is found and becomes the new lower bound

a_{n+1}

. This recursively generates two sequences

(a_n)

and

(b_n)

that narrow the interval containing the supremum.

Conditions

$M$ is a non-empty subset of $\mathbb{R}$ .
$M$ is bounded from above.
$a_1 \in M$ and $b_1$ is an upper bound for $M$ .

Reasoning, step by step

Choose an arbitrary element $a_1 \in M$ and an upper bound $b_1$ .
Calculate the midpoint $c_n = \frac{1}{2}(a_n + b_n)$ .
Check if $c_n$ is an upper bound for $M$ .
If yes, set $b_{n+1} = c_n$ and $a_{n+1} = a_n$ .
If no, find $x \in M$ such that $x > c_n$ , set $a_{n+1} = x$ and $b_{n+1} = b_n$ .
Repeat the process to generate sequences $(a_n)$ and $(b_n)$ .

Example

The video illustrates this on a number line, showing how the interval $[a_1, b_1]$ is halved at each step by evaluating the midpoint $c_1$ , updating either the lower or upper bound to get closer to the supremum of $M$ .

Common misconceptions

Assuming the method always updates both bounds simultaneously; it only updates one bound per step based on whether the midpoint is an upper bound.
Thinking the sequences $(a_n)$ and $(b_n)$ converge to different limits; they converge to the same limit, which is the supremum.

Watch the explanation

YouTubeReal Analysis 7 | Cauchy Sequences and Completeness

4:00 – 6:00Watch this moment ↗

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Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.