(related to Proposition: How Convergence Preserves Upper and Lower Bounds For Sequence Members)
This follows immediately from the corresponding rule, how convergence preserves the order relation of sequence members, if we define the constant sequences
\[l_n\le a_n\le u_n.\]
For the limits \(\lim_{n\rightarrow\infty} a_n=a\), \(\lim_{n\rightarrow\infty} l_n=L\) and \(\lim_{n\rightarrow\infty} u_n=U\) it follows then that
\[L\le a\le U.\]