Order Properties of Limits of Real Sequences

Let $(a_{n})_{n=1}^{\infty}$, $(b_{n})_{n=1}^{\infty}$ and $(c_{n})_{n=1}^{\infty}$ be sequences of real numbers, and let $L$ be a real number. Limits are as in 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 modulus of $x$ regarded as a complex number by claim 8 of Properties of Complex Conjugation and Modulus. Then the following hold.

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

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 $L$.

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 $L$.

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