Show that $$[x]+[x+\frac{1}{2}]=[2x]$$
where $$[x]$$ is the greatest integer of x

Question

perl · Answer

You can consider cases. When x is an integer, when x is not an integer

anonymous · Answer

@perl  Would you please help me out

empty · Answer

Here's a way you can solve this that I just came up with so I don't know if there's an easier way.

First let's define $0 \le d<\frac{1}{2}$

Then this means we can reach all numbers as either: $x=n+d$ or $x=n+d+\frac{1}{2}$. Why these weird choices? Cause when we plug them in we know no matter what $d$ is we have:

$$[n+d] = n$$
$$[n+d+\frac{1}{2}] = n+1$$

So we have two separate cases, plugging 'em in to:
 $$[x]+[x+\frac{1}{2}]=[2x]$$

I think that will end up working?

amoodarya · Answer

"empty' give the answer 
i give another way 
you can also take two case below 
and 
check both of them 
$$\lfloor x \rfloor =2n\ \lfloor x \rfloor=2n+1$$