Question

Describe how to construct a sequence of nested intervals that have precisely the number √2 in common. There are many possible ways to construct the sequence, just find one of them.

Answer 1

Answer:

See answer below

Step-by-step explanation:

Define the intervals:

$I_n:=(\sqrt{2}-\frac{1}{n},\sqrt{2}+1/n)$ for n≥1.

The intervals are nested, in the sense that $I_1\supset I_2\supset I_3 \supset \cdots$ To see this, use the fact that for all n≥1, -1/n≤-1/(n+1) and 1/n≥1/(n+1) (intuitively, the intervals are "shrinking" in size, and are centered around √2).

The only point common to all intervals is √2, in the sense that $\bigcap_{n\geq 1}I_n=\{2\}$  

The idea for this construction is to center the intervals around √2 and shrink their size with the summand 1/n. As n goes to infinity, 1/n tends to zero and the intervals became closer and closer to √2, but they NEVER degenerate to the point √2, in contrast to their intersection.