Proving that a function is a metric Let 
$$p(x,y)= \left|\frac{1}{x} - \frac{1}{y}\right|$$
for $x,y > 0$.
Prove that $p$ is a metric for $(0,\infty)$.
This question is from Methods of Real Analysis, 2nd edition by Richard Goldberg.
I do not have any idea how can I solve it 
Also, if anyone know where I can get this book solution manual please let me know.
 A: This is more of a hint than an answer, in the sense that I've left out the part of the proof that's not totally immediate. I hope that's all right. Let me know if you're still stuck and I'll update my answer.
The definition of a metric requires a function to be nonnegative, symmetric in its arguments, to be zero only for equal arguments, and to satisfy the triangle inequality. To prove that a function is a metric just requires proving that it has these four properties. 
Nonnegativity is easy--there's an absolute value around the whole thing.
Symmetry is easy for a similar reason. Replace $x$ by $y$ and vice versa, and the result remains the same because of the absolute value.
Seeing that $p(x,y) = 0$ if and only if $x = y$ shouldn't be too much of a challenge:
$$\frac{1}{x} - \frac{1}{y} = \frac{y-x}{xy},$$
and you should be able to take it from there.
Using the equality I showed you above, and the triangle inequality for the absolute value, you should be able to finish the rest of the proof.
I'm sorry, but I don't know about the solution manual to that textbook. Hopefully someone else can help there.
