TITLE: Is strong approximation difficult?
QUESTION [24 upvotes]: Recently a colleague and I needed to use the fact that the natural map $SL_2(\mathbb{Z}) \rightarrow SL_2(\mathbb{Z}/N\mathbb{Z})$ is surjective for each $N$. I happily chugged my way through an elementary proof, but my colleague pointed out to me that this is a consequence of strong approximation.
After browsing through some references (including Ch. 7 of Platonov and Rapinchuk, and one of Kneser's papers) I now cheerfully agree with my colleague. Indeed, as P+R describe, strong approximation is "the algebro-geometric version of the Chinese Remainder Theorem".
But what about the proofs? To this particular novice in this area, the proofs seemed rather difficult -- involving a variety of cohomology calculations, nontrivial theorems about Lie groups, and much else besides. I confess I did not attempt to read them closely.
But perhaps this is because the theorems are proved in a great deal of generality, or perhaps such arguments are regarded as routine by experts. My question is this:

Is the proof of strong approximation genuinely difficult?

For example, can a reasonably simple proof be given for $SL_n$ which does not involve too much machinery, but which illustrates the concepts behind the general case?

REPLY [8 votes]: I am quite surprised that it seems to be so poorly known, but an elementary proof of Strong Approximation for $SL_n$ over a Dedekind domain can already be found in Bourbaki (Algebre  Commutative, VII, $\S$2, n.4). Essentially, the idea is to deduce it from the Chinese Remainder Theorem, by means of elementary matrices, as hinted in Ralph's answer. Actually, I would be curious to know a bit more about the history of this case of Strong Approximation: who and when did prove it first?