Proving the Validity of a New Metric Space

[Lily@pie]

Suppose that (X,d) is a metric

Show \tilde{d}(x,y) = \frac{d(x,y)}{\sqrt{1+d(x,y)}} is also a metric

I've proven the positivity and symmetry of it.

Left to prove something like this

Given a\leqb+c
Show \frac{a}{\sqrt{1+a}}\leq\frac{b}{\sqrt{1+b}}+\frac{c}{\sqrt{1+c}}

I try to prove this
a\sqrt{1+b}\sqrt{1+c}=b\sqrt{1+a}\sqrt{1+c}+c\sqrt{1+a}\sqrt{1+b}
but I'm just stuck!

cz previously I've proven this \frac{a}{1+a}\leq\frac{b}{1+b}+\frac{c}{1+c} before...

I guess I can't use the same method??

 

[Norwegian]

Given a\leqb+c
Show \frac{a}{\sqrt{1+a}}\leq\frac{b}{\sqrt{1+b}}+\frac{c}{\sqrt{1+c}}

Can you prove this assuming a is bigger than both b and c? Can you prove it assuming a is not the biggest?

 

[Lily@pie]

What do you mean? I know a≤b+c...

So I assume a is bigger than both a and c?

 

[Norwegian]

What do you mean?

I mean that if you can prove an assertion under some condition, and then under the complementary condition, then you are done. And I have also told you what condition works well in your case.

 

[Jamma]

This is natural to assume given the triangle inequality for the original metric.

 

[Lily@pie]

Let:
a=d(x,y)
b=d(x,z)
c=d(z,y)

a\leqb+c
This is the triangle inequality for the original metric right?

Show \frac{a}{\sqrt{1+a}}\leq\frac{b}{\sqrt{1+b}}+\frac{c}{\sqrt{1+c}} -- (*)

Does this means that I try to prove (*) by two cases:
1. assuming a>b and a>c
2. assuming a<b and a<c?

 

[Lily@pie]

I'm not sure whether this way of proving it is okay:

I take a look at the function

f(x)=\frac{x}{\sqrt{1+x}}

f'(x)=\frac{2+x}{2(1+x)^{\frac{3}{2}}}
f'(x)>0 if and only if x≥-1.

Since I know that 0≤a≤b+c, f(x) is an increasing function in the interval [0,b+c]. Therefore, by choosing a=b+c, f(a)=\frac{a}{\sqrt{1+a}}=\frac{b+c}{\sqrt{1+b+c}} is the maximum possible value.

Hence it is enough to show that \frac{b+c}{\sqrt{1+b+c}}\leq\frac{b}{\sqrt{1+b}}+\frac{c}{\sqrt{1+c}}

Since 1+b+c≥1+b and 1+b+c≥1+c,
\frac{b+c}{\sqrt{1+b+c}}=\frac{b}{\sqrt{1+b+c}}+\frac{c}{\sqrt{1+b+c}}\leq\frac{b}{\sqrt{1+b}}+\frac{c}{\sqrt{1+c}}

Therefore,
\frac{a}{\sqrt{1+a}}\leq\frac{b}{\sqrt{1+b}}+\frac{c}{\sqrt{1+c}}

It seems a bit dodgy

 

[Norwegian]

Your solution is good. You use both the idea about f'>0 and the analysis of the denomintarors in one argument.

 

[Lily@pie]

So is this method valid?