Why can't we replace $\leq$ with $<$ in the limit location theorem?

Question

The limit location theorem states for a sequence $\{a_n\}$, if for large enough $n$ we have $a_n \leq M$, then $\lim_{n \to \infty} a_n \leq M$ (the same holds if we replace $\leq$ with $\geq$). Why does the theorem not hold with strict inequalities?

Answer 1

In general we have that even if $a_n < M$ then $\lim_{n \to \infty} a_n \leq M$, consider for example

$$\forall n>0 \quad a_n=\frac{n}{n+1}<1$$

but

$$\lim_{n \to \infty}a_n=\lim_{n \to \infty}\frac{n}{n+1}=1$$

thus the theorem you mentioned considers the more general case.