Actually, even if no one dies, we can still all have children without creating infinite population, as long as we have, on average, less than 2 per couple (1 per person).

For example, if we have 1 billion people who suddenly all become immortal, and each couple has, on average 1.8 children, we asymptote towards a world with 10 billion immortals. Everyone could have offspring, though the actual number of young people would be continuously decreasing...

If $p0$ is the starting population, and $a$ is the number of children per person (number per couple divided by two), we have

  p = p0 * sum_{i=0...inf} a^i = p0 / (1-a)