Here's a multi-part thought experiment.  It's not "proof" of anything in particular. But I do claim that it gives us at least some empirical support for hope about the future — and, by extension, support for trusting in God.
1. Assume you're an average person living today. Knowing what you know today, would you trade places with an average person who lived in your ancestral culture (be it Europe, Africa, Asia, or whatever) 10,000 years ago? 1,000 years? 100 years? 50 years ago?
Remember that, in these earlier eras, you wouldn't have been a king or queen. You'd have been one of the average people of the culture in question. You would have to have lived as they did. You'd have had to deal with things such as food scarcities; poor sanitation, primitive medical care, high infant mortality, short life expectancy, etc.
2. Let's take the experiment one step further. Suppose you were an average person in your ancestral culture during one of those earlier eras. Again knowing what you know today, would you have been willing to trade places with an average person of a still-earlier era? For example, if you had lived as an average Englishman during the reign of Elizabeth I, do you think you'd have been willing to swap lives with an average Angle or Saxon or Dane or Viking or Norman of 1,000 years or 10,000 years earlier?
3. Finally, do the above experiments again, but this time, assume you're doing your time-traveling as a member of a privileged elite. Ask yourself whether you'd be willing to trade places with a comparable member of an earlier privileged class.
* * *
Let's review. With your present knowledge, would you be willing to take a chance on any of these swaps with earlier eras, either average-to-average or elite-to-elite?
I bet your answer is something like, "not a *$&#@ chance, boyo."
If I'm right, that seems to say something: Over time and on average — in fits and starts, and often at terrible cost — the human race as a whole has made reasonably steady progress. This progress has been going on for tens of thousands of years, in the teeth of our entropic tendencies toward murder, war, and plain stupidity.
Our progress has not been nearly uniform. History is replete with examples of serious reverses; and when progress does happen, we don't know how to make sure that everyone benefits. We don't even have a decent grasp of why we make progress. 
Moreover, as mutual fund advertisements often say, past performance is not necessarily a good indicator of future prospects. For all we know, all of our past progress could be wiped out by self-destruction, catastrophic pandemics, asteroid strikes, or some other unpleasant event or events.
But even so, the big picture of progress is encouraging, no? It offers a reason for hope that is based on more than just a leap of faith.
 This essay draws on Gregg Easterbrook's The Progress Paradox: How Life Gets Better While People Feel Worse (2003).
 There's a case to be made that human progress is driven by a variety of particular human abilities that Christians and Jews traditionally regard as gifts from God, such as:
- the ability to learn about the universe as it is;
- the ability to imagine how the universe might be different,
- the ability to want the universe to be different in a particular way;
- the ability to envision a possible course of action to make the universe different in that particular way;
- the ability to remember and learn from the past, and to modify future actions based on experience;
- the ability to collaborate with others, including passing along useful information;
- the ability to desire the welfare of others, which helps provide the trust necessary for collaboration.
But that's another topic.