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
x-[x]<(1/2).
multiply both sides with 2,we will get
2x-2[x]<1.--->(1).
2x<1+2[x].
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).
2.2-2[2]<1.
4-4<1.
0<1.
Here,From this ,It is true that 0 is less than 1.
So,2x=2[x].
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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