Question

Prove that the converse of Theorem \(7.9\) is not true by finding a continuous function \(a:\left [ 1,\infty \right )\rightarrow R\) such that \(\lim_{x\rightarrow \infty }a\left \( x \right \)\) does not exist but \(\left \{ a\left \( x \right \) \right \}\) converges.

Short answer

An example of a continuous function \(a:\left [ 1,\infty \right )\rightarrow R\) such that \(\lim_{x\rightarrow \infty }a\left \( x \right \)\) does not exist but \(\left \{ a\left \( x \right \) \right \}\) converges is \(a\left ( x \right )=\sin \left ( \pi x \right )\).

Step-by-step solution

Step 1. Given Information

The objective is to find an example of a continuous function \(a:\left [ 1,\infty \right )\rightarrow R\) such that \(\lim_{x\rightarrow \infty }a\left \( x \right \)\) does not exist but \(\left \{ a\left \( x \right \) \right \}\) converges.

Step 2. Example

Consider the function \(a\left ( x \right )=\sin \left ( \pi x \right )\).

The function is continuous on \(\left [ 1,\infty \right )\) and \(\lim_{x\rightarrow \infty }a\left \( x \right \)\) does not exist.

CASE \(1\): \(x \epsilon \mathbb{Z}\) then

\(\begin{align*}

a\left \{ x \right \} &= \sin\left ( \pi x \right ) \\

a\left \{ x \right \} &= 0

\end{align*}\)

Then, (\lim_{x\rightarrow \infty }a\left \( x \right \)=0\).

Step 3. Explanation for next case

Case \(2\): If x=\frac{4n+1}{2}, n\epsilon \mathbb{N} then

\(\begin{align*}

a\left \{ x \right \} &= \sin\left ( \pi x \right ) \\

a\left \{ x \right \} &= \sin\left ( \frac{4n+1}{2}\pi \right ) \\

a\left \{ x \right \} &= 1

\end{align*}\)

and

\(\begin{align*}

\lim_{x\rightarrow \infty }a\left \{ x \right \} &= \lim_{x\rightarrow \infty } \sin\left ( \pi x \right ) \\

&= \lim_{x\rightarrow \infty } \sin\left ( \frac{4n+1}{2}\pi \right ) \\

&= 1

\end{align*}\)

Step 4. Explanation

So, it is clear that limit does not exist.

Now,

\(\begin{align*}

a_{k}&=a(k) \\

& =\sin\left (k\pi \right )

\end{align*}\)

As, \(k\epsilon \mathbb{N}\) so \(\sin\left (k\pi \right )=0\).

As each term of the series is \(0\), so \left \(\{ a\left ( x \right ) \right \}\) converges to \(0\).