Question

aratati ca ptr orice x,y apartin lui R avem [x]=[y] rezulta |x-1|<1​

Answer 1

[x]=[y] rezulta |x-y|<1

x=[x]+{x} si y=[y]+{y}

x-y=[x]+{x}-[y]-{y}; dar [x]=[y]

=> x-y={x}-{y}

0≤{x}<1  

0≤{y}<1 => 0≥ -{y}>-1, sau -1<-{y}≤0

=> -1<{x}-{y}<1

|{x}-{y}|<1

=> |x-y|<1