Squeeze Theorem for Real Sequences/Corollary

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

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.11$: Corollary