Proof that Bitcoin is worthless:
I set up camp over the Golden Gate Bridge. I ask people if they'd be willing to throw their lunch into the water (use their computer's time/energy/money to mine Bitcoin), with no possibility of retrieving it, and in return I offer to give them a certificate stating that they've thrown their lunch away (a Bitcoin).
Now, I see no reason why anyone would expect why these certificates would have any value. But let's suppose they do have value, and people can trade their certificates, say, for a lunch (or perhaps half a lunch). NOW, suppose I have been secretly stashing away those lunches (really it doesn't need to be in secret - you don't care what happens to your lunch after you throw it away). Then there is now objectively more value in the world than there was before I set up shop - there are the same amount of lunches in the world, but there are also now certificates that have some positive value. We've made a free lunch!
This is a contradiction. If we could create value by this silly game, we could easily make as much value as we want, and we would have solved all the world's problem.
But this should be obvious in the first place! Why on earth would you expect to be rewarded with something of value (a Bitcoin) for doing something fundamentally useless to society (mining Bitcoin). Even though mining Bitcoin comes at a cost to you, this doesn't matter - it just means you're throwing your lunch away.
Now why does the US Dollar have value? Instead of throwing away our lunches for a certificate, we basically asked the government to store our lunches in Fort Knox. Today, lots of gold and other items of value are held there. Why are they held there? There is an implicit assumption that these items of value support the US Dollar. The government knows it could not start "eating those lunches" that it's got in its reserve! It would not yield society a free lunch - it would crash the US Dollar.