Since the integral of 1/x diverges at infinity, any attacker, even with an arbitrarily small fraction of the total hash rate, will almost certainly eventually overtake the network.
Yup. To be more specific:
An attacker with a constant fraction (like 0.01%) of the network hashpower, who starts attacks and keeps up their losing attack forever, constantly adjusting timestamps to make their difficulty go up as fast as possible, _where the network hashrate (and the attacker's) increases exponentially_ overtakes with finite non-zero probability, and thus eventually overtakes.
This can be intuitively understood by observing that given exponential growth in rate the total work in history is just some large finite portion of the attacker's current hashrate... and the attacker has some low but finite odds of reaching any possible luck level.
If you do the math however, for any remotely reasonable choice of numbers the time the attacker must persist before they have a 50% chance of overtaking the results end up being insanely long, thousands and thousands of years-- to which you can reasonable answer "nodes shouldn't reorg a thousand years of blocks from their own timeline"-- making the attack pretty pointless.
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(just a brief dos attack on new systems joining the network-- until someone adds the one line of code blacklist to kill that specific attack chain. ... so N-thousand years of work that could have spent mining honestly, twarted by one line of code and achieving only an attack on a few new nodes joining the network).
This also doesn't work if hashrate isn't growing exponentially (with any exponent) for everyone. Moore's law is not a physical law, it's an observation. Hashrates will stop growing at some point or another.
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