Can we use a limit argument to prove that an infimum = some value? I want to show that the infimum of the set containing the terms of the harmonic sequence is 0. Can I simply argue that because the harmonic sequence converges to 0 then the infimum of the set containing terms of the harmonic sequence is 0? Our recursive definition of a sequence is $1/n$ where $n$ starts at n=1 and n goes to infinity. Existence of Sequence in Set of Real Numbers whose Limit is Infimum?

Not exactly; the convergence alone does not ensure that conclusion. Indeed, there is a converging sequence $(x_{n})$ with limit not $\inf_{n}x_{n}$ (what is an example?). If $x_{n} := 1/n$, then $(x_{n})$ is decreasing ; so it just so happened that $\inf_{n}x_{n} = \lim_{n}x_{n}$ (Try to prove this; could you?).