Are Metric Space Infimums Equal for Non-Empty Subsets?

[choccookies]

HI I've got this question I don't know how to do;

Let X be a metric space, and let Q,J be non-empty subsets of X. prove that

inf{dist(x,J):x is a member of Q}= inf{dist(Q,y):y is a member of J}.


I know that the dist(x,J):= inf{d(x,y)|y is a member of J}, I thought maybe if I tried to show the two infimum weren't equal and prove by contradiction, because if d is a metric then d(x,y)=d(y,x). But I'm not sure how to go about it.
Any help please?

 

[micromass]

Maybe you can prove that both are equal to $$inf\{d(x,y)~\vert~x\in J, y\in Q\}$$...

 

[choccookies]

Maybe you can prove that both are equal to $$inf\{d(x,y)~\vert~x\in J, y\in Q\}$$...

but how would i do this? using the three axioms of metrics?

 

[choccookies]

Maybe you can prove that both are equal to $$inf\{d(x,y)~\vert~x\in J, y\in Q\}$$...

could i do this:

inf{dist(x,J): x member of Q}
= inf{inf{d(x,y)|y member of J}:x member of Q}

Since we know inf{d(x,y)|x member of Q, y member of J}= d(Q,J)

Then the above is = inf{d,(Q,J)},

and then the same for the right hand side?

 

[micromass]

Yes, I believe that is correct.