exercise 18. let ai,a2,…, an, be a sequence of rational numbers. the sequence is called a cauchy sequence if for every positive real number e, there is a natural number n, such that for all m, n ? n, prove that the sequence 2482m is a cauchy sequence.
Proof that the sequence 2482m is a Cauchy sequence:
Proof:
Let’s consider a Cauchy sequence {ai} with the property that for every positive real number e, there exists a natural number N such that for all m, n > N, we have |am - an| < e.
Now, let’s examine the sequence {2482m}. For any two terms 2482m and 2482n in the sequence, we can observe:
|2482m - 2482n| = 2482|m - n|
Given e > 0, we need to find an N such that for all m, n > N, it follows that |2482m - 2482n| = 2482|m - n| < e.
To achieve this, we can choose N > 2482/e. Then, for all m, n > N, we have:
|2482m -2482n| = 2482|m - n| < 2482N < 2482(2482/e) = e.
Therefore, we have shown that for any e > 0, there exists a natural number N such that for all m, n > N, |2482m - 2482n| < e. This satisfies the definition of a Cauchy sequence. Hence, the sequence 2482m is a Cauchy sequence.