I need to show that $(X, d)$ is a metric space:
$X$ is the sequence space where its elements are sequences of real numbers, and $d : X \times X \rightarrow \mathbb{R}$ where d is defined as:
$$d(x,y) = \sum_{j=1}^\infty\frac{1}{2^j} \frac{|x_j – y_j|}{1 + |x_j – y_j|}$$
I think that I need to prove the right $\frac{|x_j – y_j|}{1 + |x_j – y_j|}$ is montone increasing but I have no clue how to start with proving the triangle inequality for this expression. Any help is appreciated.
Best Answer
Hint: First, the function $f(x)=\frac{x}{1+x}$ is monotone increasing when $x \geq 0$, which can be proved by methods like taking derivative.
Second, since $f(x)=\frac{x}{1+x}$ is monotone increasing, prove the triangle inequality holds for each component, which is $\frac{|x_j - y_j|}{1 + |x_j - y_j|}+\frac{|y_j-z_j|}{1 + |y_j-z_j|} \geq \frac{|x_j-z_j|}{1 + |x_j-z_j|}$.
Third, prove the metric satisfies the triangle inequality.