Question

Prove that a sequence is bounded if and only if it is both bounded above and bounded below

Answer 1

Let $<$a_n$>$ be a sequence. given that $<$a_n$>$ is bounded, then by definition of boundedness

$\exist$ k $\in$ R such that |a_n| $\leq$ k $\forall$ n

$\implies$ -k $\leq$ |a_n| $\leq$ k $\forall$ n

$\implies$ -k $\leq$ |a_n| & |a_n| $\leq$ k $\forall$ n

which implies that $<$a_n$>$ is bounded below by -k and bounded above by k.

therefore, $<$a_n$>$ is both bounded above and bounded below.

Conversely:

now suppose that $<$a_n$>$ is both bounded above and bounded below.

let L, k $\isin$ R and $<$a_n$>$ is bounded below by L and bounded above by k for every n

$\implies$ a_n $\leq$ k & a_n$\geq$ L $\forall$ n

$\because$ -|L| - |k| $\leq$ L & k$\leq$ |L| + |k|

then,

if L $\leq$ a_n $\leq$ k $\forall$ n

$\implies$ -|L| - |k| $\leq$ L $\leq$ |a_n| $\leq$ k $\leq$ |L| + |k| $\forall$ n

$\implies$ -|L| - |k| $\leq$ |a_n| $\leq$ |L| + |k| $\forall$ n

hence we conclude that $<$a_n$>$ is bounded by |L| + |k|.

$<$a_n$>$ is bounded.