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Chapter 3: Problem 4

For any \(b \in \mathbb{R}\), prove that \(\lim (b / n)=0\).

Short Answer

Expert verified
The limit as n tends to infinity of the sequence \(b/n\) is 0 for any real number b.

Step by step solution

01

Identify the Sequence

The sequence \(a_n\) is given by \(a_n = b/n\) for every natural number n and any real number b.
02

Apply Limit Definition

For the sequence \(a_n\), we can now analyze the limit as n tends to infinity. This is represented as \(\lim_{n \to \infty} a_n = \lim_{n \to \infty} b/n\).
03

Compute Limit

As n increases without bound in the denominator, the fraction \(b/n\) tends towards 0. Therefore, \(\lim_{n \to \infty} b/n = 0\).

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Most popular questions from this chapter

If \(\lim \left(x_{n}\right)=x>0\), show that there exists a natural number \(K\) such that if \(n \geq K\), then \(\frac{1}{2} x<\) \(x_{n}<2 x\)

Show that \(\lim \left(1 / 3^{n}\right)=0\).

Let \(b \in \mathbb{R}\) satisfy \(0

Show that the following sequences are divergent. (a) \(\left(1-(-1)^{n}+1 / n\right)\), (b) \((\sin n \pi / 4)\).

Give an example of a bounded sequence that is not a Cauchy sequence. i \(\cdots\)
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