>its been proven that its cheaper to give the homeless housing than to pay for the resulting issues of not providing housing
is misleading and inaccurate. First, you could never prove something like this as the comparison is highly dependent upon the larger institutional framework a particular society is operating under. Second, even if it were demonstrably true in a given community at a particular point in time, there is no guarantee those same conditions would persist over time. That's not to say there aren't good arguments for providing housing or other forms of welfare, but we don't need to engage in overly sweeping or simplistic arguments to make that case.