If the cost to secure the network tends to zero, then the cost to attack the network tends to zero is well.
The is the essence of the "nothing-at-stake" problem with PoS that, despite claims to the contrary, remains unsolved (and IMO very likely unsolvable).
In PoS every currency unit is a small mining rig, and they are all created at Genesis block and it costs nothing. You don't have to make new regular capital investments into hardware, at least not significant ones, as for PoS any old computer will do that can run a wallet and the currency units inside the wallet do the mining. So the cost of securing the network is minimal, not zero, but perhaps 1-2% of the cost of the PoW network.
Cost of the attack of the PoS network is larger than cost of the attack on PoW network, as it's easier to buy hardware than to buy 51% currency units in existance in a PoS system, as that would drive the price billions and billions high.
No one has proven that it is possible to solve the "nothing-at-stake" problem. In fact it is becoming clear that any PoX-type system can be attacked if the attacker holds 51% of the critical resource X (X = W for PoW and X = S for PoS). It is easy to calculate the cost of an attack on PoW (it is the cost to acquire 51% of the hash power). But what is the cost to attack a PoS system?
Do you understand the "nothing-at-stake" problem (aka the "history re-write" problem)?