# sequence $a_n = \lceil \sqrt{2}n \rceil$

I was trying to prove $\lceil \sqrt{2}n \rceil + \lceil \sqrt{2}m \rceil \geq \lceil \sqrt{2}(n+m) \rceil$ where $m,n\in \mathbb{z}$

Direct proof I tried but could not figure out. I tried fixing m and induction on n, for n = 1 holds,

then i got stucked for k+1 case.

Is there anything fancy required here which I may have to consider?

Thanks.

Simply show that for all real numbers $a$ and $b$, we have
$$\lceil a \rceil + \lceil b \rceil \geq \lceil a+b \rceil.$$
Hint for a clever (but unnecessarily difficult) solution: consider a sequence of cleverly chosen rational numbers greater than $\sqrt{2}$ that converge and then prove the inequality. For those numbers.