Minimum Rule for Real Sequences

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Theorem

Let $\sequence {x_n}$ and $\sequence {y_n}$ be sequences in $\R$.

Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent to the following limits:

$\displaystyle \lim_{n \mathop \to \infty} x_n = l$
$\displaystyle \lim_{n \mathop \to \infty} y_n = m$


Then:

$\displaystyle \lim_{n \mathop \to \infty} \min \set {x_n, y_n} = \min \set {l, m}$


Proof

By Sum Less Maximum is Minimum:

$\forall n \in \R: \min \set {x_n, y_n} = x_n + y_n - \max \set {x_n, y_n}$

and

$\min \set {m, l} = m + l - \max \set {m, l}$


By Maximum Rule for Real Sequences:

$\displaystyle \lim_{n \mathop \to \infty} \max \set {x_n, y_n} = \max \set {m, l}$

By the Multiple Rule for Real Sequences:

$\displaystyle \lim_{n \mathop \to \infty} - \max \set {x_n, y_n} = - \max \set {m, l}$

By the Sum Rule for Real Sequences:

$\displaystyle \lim_{n \mathop \to \infty} x_n + y_n - \max \set {x_n, y_n} = m + l - \max \set {m, l}$


Hence:

$\displaystyle \lim_{n \mathop \to \infty} \min \set {x_n, y_n} = \min \set {l, m}$

$\blacksquare$


Also see