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Question Number 168628 by SANOGO last updated on 14/Apr/22

montrer que d(x,y)=((∣u−v∣)/(1+∣u−v∣)) une distance sur R

montrerqued(x,y)=∣u−v∣1+∣u−v∣unedistancesurR

Answered by ArielVyny last updated on 15/Apr/22

d(x,y)=((∣x−y∣)/(1+∣x−y∣)) si d est une distance alors −d(x,y)=d(y,x) ainsi, d(x,y)=((∣x−y∣)/(1+∣x−y∣)) et d(y,x)=((∣y−x∣)/(1+∣y−x∣)) or ∣y−x∣=∣x−y∣ ainsi d(x,y)=d(y,x) −d(x,x)≥0 ce qui est immediat car ((∣x−y∣)/(1+∣x−y∣))≥0 si d(x,y)=0→x=y d(x,x)=((∣x−x∣)/(1+∣x−x∣))=0

d(x,y)=∣x−y∣1+∣x−y∣sidestunedistancealors−d(x,y)=d(y,x)ainsi,d(x,y)=∣x−y∣1+∣x−y∣etd(y,x)=∣y−x∣1+∣y−x∣or∣y−x∣=∣x−y∣ainsid(x,y)=d(y,x)−d(x,x)⩾0cequiestimmediatcar∣x−y∣1+∣x−y∣⩾0sid(x,y)=0→x=yd(x,x)=∣x−x∣1+∣x−x∣=0

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