Squeeze Theorem for Real Sequences/Corollary

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Corollary to Squeeze Theorem for Real Sequences

Let $\sequence {y_n}$ be a sequence in $\R$ which is null, that is:

$y_n \to 0$ as $n \to \infty$

Let:

$\forall n \in \N: \size {x_n - l} \le y_n$


Then $x_n \to l$ as $n \to \infty$.


Proof

From Negative of Absolute Value: Corollary 2:

$\size {x_n - l} \le y_n \iff l - y_n \le x_n \le l + y_n$

From the Difference Rule for Real Sequences:

$l - y_n \to l$ as $n \to \infty$

and from the Sum Rule for Real Sequences:

$l + y_n \to l$ as $n \to \infty$


So by the Squeeze Theorem for Real Sequences, $x_n \to l$ as $n \to \infty$.

$\blacksquare$


Sources