Results Concerning Limits of Sequences of Intervals

Decreasing Sequences of Sets

Limit of Decreasing Sequence of Unbounded Below Closed Intervals with Endpoint Tending to Negative Infinity

Let $\sequence {x_n}_{n \mathop \in \N}$ be a decreasing sequence with $x_n \to -\infty$.

Then:

$\ds \bigcap_{n \mathop = 1}^\infty \hointl {-\infty} {x_n} = \O$

That is:

$\hointl {-\infty} {x_n} \downarrow \O$

where $\downarrow$ denotes the limit of decreasing sequence of sets.

Limit of Decreasing Sequence of Unbounded Below Closed Intervals

Let $x \in \R$.

Let $\sequence {x_n}_{n \mathop \in \N}$ be a decreasing sequence converging to $x$.

Then:

$\ds \bigcap_{n \mathop = 1}^\infty \hointl {-\infty} {x_n} = \hointl {-\infty} x$

That is:

$\hointl {-\infty} {x_n} \downarrow \hointl {-\infty} x$

where $\downarrow$ denotes the limit of decreasing sequence of sets.

Limit of Decreasing Sequence of Left Half-Open Intervals with Lower Bound Converging to Upper Bound

Let $a, b \in \R$ have $a < b$.

Let $\sequence {a_n}_{n \mathop \in \N}$ be an increasing sequence with $a_n \to b$.

Then we have:

$\ds \bigcap_{n \mathop = 1}^\infty \hointl {a_n} b = \set b$

That is:

$\hointl {a_n} b \downarrow \set b$

where $\downarrow$ denotes the limit of decreasing sequence of sets.