I looked at some rigorous definitions of limit of a function at infinity (in real analysis). The one I found most often is this:
I will assume the +∞ case below.
I haven't been able to figure out why is it required that the function be defined in an interval (a,∞).
Won't it be sufficient just to say that for any c∈ℝ, there exists m∈ℝ, m>c such that the function is defined at m (the domain of the function has no upper bound)? Such definition will be less restrictive and can directly define limit of a sequence. Is there something wrong with that definition?