In class, we used the fact that $\lceil{a + b \rceil} \geq \lceil{a}\rceil + \lfloor{b}\rfloor$. However, we weren't given a proof of this statement.
I am interested to see how this works. Can anyone help? Thanks!
In class, we used the fact that $\lceil{a + b \rceil} \geq \lceil{a}\rceil + \lfloor{b}\rfloor$. However, we weren't given a proof of this statement.
I am interested to see how this works. Can anyone help? Thanks!
By definition: $b\geq \lfloor b\rfloor$ Adding $a$ to both sides: $a+b \geq a+ \lfloor b\rfloor$ Taking the ceiling of both sides: $\lceil a + b\rceil \geq \lceil a + \lfloor b\rfloor\rceil = \lceil a\rceil + \lfloor b \rfloor$
This uses that if $n$ is an integer, then $\lceil a + n\rceil = \lceil a \rceil + n$ And if $x\geq y$ then $\lceil x \rceil \geq \lceil y\rceil$
By subtracting off integer parts, we can prove this for numbers in $[0,1)$. Unless both are $0$ the right side is $1$, and then the left is at least $1$.