Question

Prove the statement

For all real numbers x, if x − ⌊x⌋ < 1/2 then ⌊2x⌋ = 2⌊x⌋.

Answer #1

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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