Yeah, it's true. Our way of life is entirely unsustainable, and I can prove it with a few very simple arguments.
There's a finite amount of energy available on our planet. Almost all of it comes from the sun, which is converted by plants into stems and leaves and other good stuff they use to survive. Herbivores eat the plants and steal some of the energy within them, which in turn helps them grow and develop. Finally, carnivores eat the herbivores and steal their energy. We're pretty much at the top of this pyramid, enjoying the benefits of consuming hundreds of other organisms. There is less energy available to each level as you ascend this structure, however, because the transfer of energy is not entirely efficient. Therefore, there are far less carnivores than plants, for example.
It used to be that we were limited by this cycle of birth and death and the energy available within our own lifetimes, but then we discovered fossil fuels. These fuels are formed from the remains of dead plants and animals, crushed under the weight of the earth and water for millions of years. Because of this, they contain a portion of the sun's energy gathered throughout the entire history of life on our planet. After we found this source of (seemingly) limitless energy, we started making machines that could accomplish what no organism limited by photosynthesis could do. This made our lives pretty easy, and since we had so much extra energy in the form of food, our population began to explode. Things were looking pretty good for quite a while.
We can maintain our current lifestyles only because fossil fuels continue to be extracted from the earth, but there is a huge, glaring problem with this. It takes so long for fossil fuels to form that they are not being replaced. When they run out, the energy is gone for good, and we'll be forced to return to a simpler way of life where we depend on the sun for the majority of our energy. This is the first and most obvious way that our civilization is unsustainable.
Fossil fuels, unfortunately, also contain a lot of toxins that have been built up over the millenia, and burning them releases these into the air where they can travel pretty much everywhere. This has caused considerable pollution of the water, air, and soil, as well as living things that depend on them. Countries that do not have access to oil and natural gas tend to use plants to get their energy by burning wood. We also use it for cooking, construction, and for refining metals, among other things. The world's forests are disappearing at a rate that will make most of them a memory by the end of this century, simultaneously killing off many of the creatures that call them home. By the time we run out of fossil fuels (and it doesn't matter when that happens), we won't even have a healthy natural environment to fall back on. This will make our transition back to the pre-industrial age even more difficult, if not impossible. How will we purify our water? How will we grow our food? Both are accomplished using fossil fuels. This is the second way that our civilization is unsustainable.
There's one more problem we need to look at, and it relates to the first line I wrote about the finite amount of energy on our planet. If we ignore the added energy of fossil fuels, there is little doubt that the available energy from the sun, via plants, has decreased significantly for about 200 years. Therefore, the total biomass of the earth must also decrease, because there just isn't enough energy left to support every living thing. The only way it can remain the same is for organisms to evolve greater efficiency of energy transfer. We can rule that out because evolution simply takes too long to adapt in the time frames we are currently dealing with. Well, that means that some critters need to die, and typically that happens near the top of the pyramid. Remember who's up there? Yeah... us. Don't worry, though. We're pretty adaptable after all, so let's assume that we're not the ones who will start croaking. That leaves all the other animals below us, many of whom are already extinct thanks to our efforts to sustain ourselves. Essentially, the only way for our population to stay at its current value (and remember that it will likely increase to 10 billion or more) is for many more creatures to go extinct. In the end, it might end up being just plants, insects, a couple of hearty animals, and humans. I don't know about you, but that's not a world I want to live in, thanks. This is the final way that our civilization is unsustainable.
In summation, the population needs to come WAY down, and we need to stop using fossil fuels NOW! You'll probably have to give up your IPhone, your computer, your BBQ, and a few other things, but the alternative is just not worth it. Your grandchildren are gonna be pissed off at you.