I have a list of $n$ items and would like to randomly select an $m$ set from it efficiently (in terms of time complexity). Also, I want all possible subsets to be selected with equal probability. The obvious solution is to pick a random integer from $1$ to $n$ and choose the corresponding element, then repeat $m$ times, not counting the event in which one chooses and already chosen element. This becomes increasingly inefficient as $m$ approaches $n$ so for $m>n/2$ it would make sense to instead to pick a $(n-m)$-set and return its compliment.
For values of $m$ close to $n/2$, a better solution I think would be to consider each of the $n$ elements and decide either to pick that element or discard it, each time updating the probability of picking or discarding depending on the number of elements chosen vs discarded prior. Specifically, the algorithm would go as follows (python):
def randomSubset(n,m): L =  for i in range(n): if uniform(0,1)<m/(n-i): L,m = L+[i],m-1 return L
However I am concerned that this may not result in each subset being chosen with equal probability.
I have two questions. First, does this algorithm pick subsets with equal probability (if so, I'd like a proof that it does and if not I'd also like a proof that it doesn't). Second, more broadly I would like to know what good solutions exist to this problem. Clearly if $m<<n$ then the first method is better than the second however at some point the second method (if it does in fact work) is better than the first. Moreover, an entirely different approach may be best in general.