How do I show this sequence is unbounded.
${b_j=j}$ from j=1 to infinity
By using the following definition.
${b_j}$ is called bounded if there exist $M>0$
such that $b_j<M$ for all $j\in$ Natural numbers.
So the sequence is
$1,2,3,4,5,……$
It seem obvious that it is unbounded because not matter what you pick for M
there will be an integer bigger than it.
But how do you show this?
Best Answer
To prove that the sequence defined by $b_j = j$ is unbounded, let $M$ be an arbitrary natural number and observe that $M \not> M + 1 = b_{M+1}$ so that $M$ is not an upper bound of the $b_j$.