Why does this sort algorithm work? The following O(n^2) sorting algorithm works but I can't figure out why.

array a // length n, indexed 0 to n-1 

for ( i = 0; i &lt; n; i++)
   for ( j = 0; j &lt; n; j++)
       if(a[i]&lt;=a[j])
            swap(a[i],a[j]);


It seems it is not bubble sort as after a single iteration neither the minimum or maximum is in place. (And also it needs n^2 comparisons instead of n*(n-1)/2)

How would you prove this algorithm sorts ? How is this sort algorithm called?

An execution of the sort for the example :

initial target
194600925
592040703
20865352
814014281
792862803

target after iteration 0
814014281
194600925
20865352
592040703
792862803

target after iteration 1
194600925
814014281
20865352
592040703
792862803

target after iteration 2
20865352
194600925
814014281
592040703
792862803

target after iteration 3
20865352
194600925
592040703
814014281
792862803

target after iteration 4
20865352
194600925
592040703
792862803
814014281
It is a variant of bubble sort, however the endpoints of the array shift throughout the progress of the algorithm. 

In particular it maintains the following invariant:


  at the end of the $i$-th iteration, the elements in indices $1..i$ are sorted. 


At the first step it finds the maximal element and puts it in index 1.
(prefix length =1). Then, at every iteration $i$ it adds the element in position $i$ to the prefix, and performs a (degenerate) bubble sort on the first $i$ indices. This bubble sort takes only one step since except for the one new element, the other $i-1$ elements are already sorted. The maximal element will now be at position $i$. The length of the sorted prefix keeps increasing at every iteration until the entire array is sorted.

I don't know if this sorting approach has a name. It looks like a type of insertion sort.