Let $(a_{n})_{n=1}^{\infty}$, $(b_{n})_{n=1}^{\infty}$ and $(c_{n})_{n=1}^{\infty}$ be \reftext{def:sequence-in-set-2026a}{sequences} of \reftext{def:real-numbers-c54-2026c}{real numbers}, and let $L$ be a real number. Limits are as in \reftext{def:limit-sequence-real-c54-2026a}{that definition}, and $|x|$ denotes the absolute value of a real number $x$, that is, $x$ if $0\le x$ and $-x$ otherwise, which coincides with the \reftext{def:complex-modulus-2026a}{modulus} of $x$ regarded as a complex number by claim 8 of \ref{lem:complex-conjugate-modulus-properties-2026a}. Then the following hold.

\textbf{1. (Comparison)} If $(a_{n})$ converges to $A$, $(b_{n})$ converges to $B$, and $a_{n}\le b_{n}$ for every $n$, then $A\le B$.

\textbf{2. (Squeeze)} If $a_{n}\le c_{n}\le b_{n}$ for every $n$ and both $(a_{n})$ and $(b_{n})$ converge to $L$, then $(c_{n})$ converges to $L$.

\textbf{3. (Domination by a null sequence)} If $|c_{n}-L|\le b_{n}$ for every $n$ and $(b_{n})$ converges to $0$, then $(c_{n})$ converges to $L$.

\textbf{4. (Absolute values)} If $(a_{n})$ converges to $A$, then $(|a_{n}|)$ converges to $|A|$.