MATH.21:5.1 - Obtain a real object from rational intervals

The task is to construct a positive number r with r²=2 and obtain rational approximations with a chosen absolute error.

Start with l_0=1 and u_0=2. At each step take the rational midpoint m. If m²≤2, replace the lower endpoint by m; otherwise replace the upper endpoint. Squaring is increasing on the positive interval, so each step retains l_n²≤2≤u_n². The intervals are nested and their widths are 2⁻ⁿ.

The lower endpoints form a bounded increasing sequence. In the real numbers, their supremum r exists. Each l_n≤r≤u_n: every later lower endpoint is at most u_n, and earlier ones are no larger than l_n. The shrinking width makes this r the only common point.

Both r² and 2 lie between l_n² and u_n². Since the endpoints stay between 1 and 2, the width of this squared interval is (u_n-l_n)(u_n+l_n)≤4·2⁻ⁿ. It tends to zero, so r²=2. This obtains the desired object without presupposing a square-root value to drive the construction.

After four bisections the interval is [22/16,23/16]. Its midpoint 45/32 differs from r by at most 1/32. For a smaller tolerance epsilon, choose a stage with 2⁻ⁿ⁻¹≤epsilon, or refine until the interval width is at most twice epsilon.

Changed space. If the result must remain rational, existence fails. In a fraction p/q in lowest terms, p²=2q² would make p even, and then q even, contradicting lowest terms. The rational approximations and their widths remain available, but they construct a real number rather than a missing rational solution. The next decision concerns admitting that extension or retaining a finite rational answer.