How to show that this function is bijective 
Possible Duplicate:
Proving the Cantor Pairing Function Bijective 

Assume I define
$$ f: \mathbb N \times \mathbb N \to \mathbb N, (a,b) \mapsto a + \frac{(a + b ) ( a + b + 1)}{2} $$
How to show that this function is bijective? For injectivity I tried to show that if $f(a,b) = f(n,m) $ then $(a,b) = (n,m)$ but I end up getting something like $3(n-a) + (n+m)^2 -(a+b)^2 + m - b = 0$ and don't see how to proceed from there. There has to be something cleverer than treating all possible cases of $a \leq n, b \leq m$ etc. 
For surjectivity I'm just stuck. If I do $f(0,n)$ and $f(n,0)$ it doesn't seem to lead anywhere.
Thanks for your help.

The term $(a+b)(a+b+1)/2$ is the sum of the numbers from $1$ to $a+b$. For a fixed value of $s=a+b$, $a$ ranges from $0$ to $s$, so we need $s+1$ different results for these arguments. Now you can prove by induction that the range from $s(s+1)/2$ to $(s+1)(s+2)/2-1$ contains precisely the images of the pairs with sum $s$.
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