Prove that $\left\lceil \frac{n}{m} \right\rceil =\left \lfloor \frac{n+m-1}{m} \right\rfloor$ On a discrete mathematics past paper, I must prove that 
$$\left\lceil \frac{n}{m} \right\rceil = \left\lfloor \frac{n+m-1}{m} \right\rfloor$$  
for all integers $n$ and all positive integers $m$.
I have started a proof by induction thus:
Let $P(m,n)$ be the statement that 
$$\left\lceil \frac{n}{m} \right\rceil = \left\lfloor \frac{n+m-1}{m} \right\rfloor$$  
for all integers $n$ and all positive integers $m$.
Then, according to the technique described here, I must prove the following:


*

*Base case: I have already proved that $P(a,b)$, where $a=1$ and where $b$ is the smallest value where $n$ is valid.  (Note that this is equivalent to proving that $\lceil n \rceil = \lfloor n \rfloor \space \forall\space n\in\mathbb{Z}.$)

*Induction over $m$: I must show that $P(k,b) \implies P(k+1,b)$ for some positive integer $k$.  THIS IS THE STAGE AT WHICH I AM STUCK.

*Induction over $n$: I must show that $P(h,k) \implies P(h,k+1)$ for some positive integer $m$ and for some integer $k$ (I think that I must account for the fact that $k$ could be negative OR non-negative).


I will explain why I am stuck.
My inductive hypothesis is that $P(k,b)$ - that is, $\left\lceil \frac{b}{k}\right\rceil = \left\lfloor \frac{b+k-1}{k}\right\rfloor$.
I want to show that this implies $P(k+1,b)$.  I have tried to do this by attempting to express $\lceil \frac{b}{k+1} \rceil$ in terms of $\left\lceil \frac{b}{k} \right\rceil$, but have not succeeded.
Any hints would be appreciated.
 A: There exist integers $N,a$ such that 
$$\frac{n}{m}=N+\frac{a}{m}\ \ \ \text{and}\ \ \ 1\le a\le m.$$
Then, the LHS equals
$$\left\lceil\frac nm\right\rceil=N+1.$$
And the RHS equals
$$\begin{align}\left\lfloor\frac{n+m-1}{m}\right\rfloor&=\left\lfloor\frac nm+1-\frac 1m\right\rfloor\\&=\left\lfloor N+1+\frac{a-1}{m}\right\rfloor\\&=N+1+\left\lfloor\frac{a-1}{m}\right\rfloor\\&=N+1.\end{align}$$
A: Hint:
Let $k$ be the (unique) integer with: $n=km+r$ with $r\in\left\{ 0,1,\dots,m-1\right\} $.
Then $\lceil\frac{n}{m}\rceil=k+\lceil\frac{r}{m}\rceil$ and $\lfloor\frac{n+m-1}{m}\rfloor=k+1+\lfloor\frac{r-1}{m}\rfloor$.
So it remains to be shown that $\lceil\frac{r}{m}\rceil=1+\lfloor\frac{r-1}{m}\rfloor$
for $r\in\left\{ 0,1,\dots,m-1\right\} $.
Discern the cases $r=0$ and $r\in\left\{ 1,\dots,m-1\right\} $.
A: We have
$$
\lfloor \frac{n+m-1}{m}\rfloor=\lfloor \frac{n-1}m \rfloor+1
$$
Now you have two cases $m|n$ or not...
If $m|n$, then $\lceil \frac nm\rceil=\frac nm$ and $\lfloor\frac{n-1}m\rfloor=\frac nm-1$
Now if $m$ does not devide $n$, then $\lceil \frac nm\rceil=\lfloor\frac nm\rfloor+1$ and $\lfloor\frac nm\rfloor=\lfloor\frac{n-1}m\rfloor$.
A: Quick approach: We have, by the division algorithm, that:
$$n=mq+r; \, q,r\in\mathbb Z; \, 0\leq r<m$$
Show that $$\left\lceil \frac{n}{m}\right\rceil = \begin{cases}q&\text{if $r=0$}\\
q+1&\text{if $r\neq 0$}\end{cases}$$
Now, if $m>r>0$ then $m>r\geq 1$ so $2m-1>r+m-1\geq m$ and deduce that $$q+1\leq\frac{n+m-1}{m}<q+2$$
