Limit of Bounded Convergent Sequence is Bounded

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Theorem

Let $\left \langle {x_n} \right \rangle$, $\left \langle {a_n} \right \rangle$, and $\left \langle {b_n} \right \rangle$ be convergent sequences in $\R$.

Let $\left \langle {x_n} \right \rangle$, $\left \langle {a_n} \right \rangle$, and $\left \langle {b_n} \right \rangle$ converge to $x, a, b \in \R$, respectively.

Suppose that:

$\exists N \in \N : n \geq N \implies a_n \leq x_n \leq b_n$

Then:

$ a \le x \le b$


Proof

Aiming for a contradiction, suppose that $x < a$.

Let $\epsilon = \dfrac{a - x} 2 > 0$

From the convergence of $\left \langle {x_n} \right \rangle$:

$\exists M_1 \in \N : n \ge M \implies x - \epsilon < x_n < x + \epsilon$

Or, equivalently:

$\exists M_1 \in \N : n \ge M \implies \dfrac {3x - a} 2 < x_n < \dfrac{x + a} 2$

From the convergence of $\left \langle {a_n} \right \rangle$:

$\exists M_2 \in \N : n \ge M \implies a - \epsilon < a_n < a + \epsilon$

Or, equivalently:

$\exists M_2 \in \N : n \ge M \implies \dfrac {x + a} 2 < a_n < \dfrac{3a - x} 2$

Let $M = \max\left\{ {N, M_1, M_2} \right\}$

Then, for any $n \ge M$:

\(\displaystyle x_n\) \(<\) \(\displaystyle \dfrac {x + a} 2\)
\(\displaystyle \) \(<\) \(\displaystyle a_n\)

This contradicts the hypothesis that:

$\forall n \ge N : a_n \le x_n$


The same argument, mutatis mutandis, brings us to a contradiction if we suppose $x > b$.

Hence the result, by Proof by Contradiction.

$\blacksquare$