Prove the statement
For all real numbers x, if x − ⌊x⌋ < 1/2 then ⌊2x⌋ = 2⌊x⌋.
Here,In the problem,x is real number.
First of all,Real number is an integer of national values which does not contain complex values in j form.
Here,Given condition is
multiply both sides with 2,we will get
Any value is definitely less than the value obtained by adding one to that value.
Now,For example take the real value of x=2.
Substitute x value in the equation (1).
Here,From this ,It is true that 0 is less than 1.
That means [2x]=2[x].
From this,[2x]-2[x]=0 is less than 1 only.
If we are taking negative real values also
[2(-2)]=2[-2]=-4 only and difference is 0 less than 1.
So,Finally if x-[x]<(1/2) ,then definitely [2x]=2[x].
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