The TLDR for how a bond that continues to pay interest forever can be valued at less than infinity dollars is due to the "time value of money", which states that $X in the future is worth less than $X today. This makes sense intuitively if you consider that if you had that money today, you could invest it and earn interest on it.<p>So since money in your hands is worth more than that same amount of money in the future, you can actually calculate how much a future cash flow is worth today by discounting it to its present value ("discounted cash flow" aka DCF).<p>To bring it back to perpetual bonds, if you DCF all of the future cash flows to their present value, you actually get a finite number (due to the diminishing nature of the cash flows that are further and further in the future).<p>For those who want to learn this in more detail, I recommend MIT's OCW course "Finance Theory I" with Andrew Lo.