Lower and Upper Bounds for Sequences

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Theorem

Let $\sequence {x_n}$ be a sequence in $\R$.

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


Then:

$(1): \quad \forall n \in \N: x_n \ge a \implies l \ge a$
$(2): \quad \forall n \in \N: x_n \le b \implies l \le b$


Corollary

Let $\left\langle{y_n}\right\rangle$ be another sequence in $\R$.

Let $y_n \to m$ as $n \to \infty$.

Suppose that for all $n \in \N$, $x_n \le y_n$.


Then $l \le m$, that is, $\displaystyle \lim_{n\to\infty} x_n \le \lim_{n\to\infty} y_n$.

This is often phrased as: limits preserve inequalities.


Proof

$(1): \quad \forall n \in \N: x_n \ge a \implies l \ge a$:

Let $\epsilon > 0$.

Then:

$\exists N \in \N: n > N \implies \left|{x_n - l}\right| < \epsilon$

So from Negative of Absolute Value:

$l - \epsilon < x_n < l + \epsilon$

But $x_n \ge a$, so:

$a \le x_n < l + \epsilon$

Thus, for any $\epsilon > 0$:

$a < l + \epsilon$

From Real Plus Epsilon it follows that $a \le l$.

$\Box$


$(2): \quad \forall n \in \N: x_n \le b \implies l \le b$:

If $x_n \le b$ it follows that $-x_n \ge -b$ and the above result can be used.

$\blacksquare$


Warning

Let $\sequence {x_n}$ be a sequence in $\R$.

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


Then it is not the case that:

$(1): \quad \forall n \in \N: x_n > a \implies l > a$
$(2): \quad \forall n \in \N: x_n < b \implies l < b$


Sources