- #1

mynameisfunk

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Prove or disprove d is a metrix in [tex]X[/tex]:

[tex]d(x,y)=|x^3-y^3|[/tex]

OK, 3 conditions to meet:

(i) [tex]d(x,y)>0[/tex]

(ii) [tex]d(x,y)=0 [/tex] iff [tex] x=y[/tex]

(iii) [tex]d(x,y) \leq d(x,r) + d(r,y)[/tex] for any [tex]r \epsilon X[/tex]

the first 2 are obvious and I have solved this by proving all of the cases:

[tex]r \leq x \leq y[/tex], [tex]x \leq r \leq y[/tex], etc.

My problem is that I know there is a better proof that is much shorter. My professor did it in class but I still had gotten the problem correct so didnt write it down. Any suggestions on a simpler way to do this?

[tex]d(x,y)=|x^3-y^3|[/tex]

OK, 3 conditions to meet:

(i) [tex]d(x,y)>0[/tex]

(ii) [tex]d(x,y)=0 [/tex] iff [tex] x=y[/tex]

(iii) [tex]d(x,y) \leq d(x,r) + d(r,y)[/tex] for any [tex]r \epsilon X[/tex]

the first 2 are obvious and I have solved this by proving all of the cases:

[tex]r \leq x \leq y[/tex], [tex]x \leq r \leq y[/tex], etc.

My problem is that I know there is a better proof that is much shorter. My professor did it in class but I still had gotten the problem correct so didnt write it down. Any suggestions on a simpler way to do this?

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