  When verifying computer systems we sometimes want to study their asymptotic
behaviors, i.e., how they behave in the long run. In such cases, we need real
analysis, the area of mathematics that deals with limits and the foundations of
calculus. In a prior work, we used real analysis in ACL2s to study the
asymptotic behavior of the RTO computation, commonly used in congestion control
algorithms across the Internet. One key component in our RTO computation
analysis was proving in ACL2s that for all alpha in [0, 1), the limit as n
approaches infinity of alpha raised to n is zero. Whereas the most obvious
proof strategy involves the logarithm, whose codomain includes irrationals, by
default ACL2 only supports rationals, which forced us to take a non-standard
approach. In this paper, we explore different approaches to proving the above
result in ACL2(r) and ACL2s, from the perspective of a relatively new user to
each. We also contextualize the theorem by showing how it allowed us to prove
important asymptotic properties of the RTO computation. Finally, we discuss
tradeoffs between the various proof strategies and directions for future
research.
