Squeeze Theorem for Sequences/Corollary

Corollary to Squeeze Theorem for Sequences

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

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

Let:

$\forall n \in \N: \left|{x_n - l}\right| \le y_n$

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

Proof

From the corollary to Negative of Absolute Value, we have:

$\left|{x_n - l}\right| \le y_n \iff l - y_n \le x_n \le l + y_n$

From the Combination Theorem for Sequences: Sum Rule:

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

and:

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

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

$\blacksquare$

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 4.11$