I'm taking my first calculus courses and I am just being introduced to such concepts as $\limsup$ and $\liminf$ , so from what I understood if we consider some convergent real sequence then all it's subsequences converge to the same limit, meaning $\limsup$ and $\liminf$ of this sequence do not exist? is this right? please correct me if I am wrong.
2026-04-08 04:17:20.1775621840
Lim Sup and Lim Inf notions
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If a sequence is convergent, then both {limes superior and limes inferior exists and are equal ( to the limit of the sequence ).
The interesting case is when a sequence has more than one accumulation points. In such cases limes superior is the biggest (if there is such) of the accumulation points and limes inferior is the smallest (if there is such). For example for $a_n = (-1)^n $ there are just two accumulation points, namely $-1,1$. Thus $\limsup a_n = 1$ and $\liminf a_n = -1$.
In fact exactly this property is what characterizes the convergent sequences ( in terms of limes superior and limes inferior) You can check out this , for example ,sequence converges iff limsup=liminf