Let Xn and Yn be two series of real numbers such that Xn<Yn for all n>N.
Show that a)lim inf Xn<lim inf Yn b)lim supXn<lim sup Yn
Both inequalities are false: $\displaystyle \frac{n-1}{n}<\frac{n+1}{n}\,\,\,\forall\,\,n\in\mathbb{N}$, but there's equality in (a) and in (b) since the limit exists in both cases and is thus equal to the lower and upper limits.