This question was asked in our math test. What I did was assume that a divergent sequence is not bounded below. Then we would not know if the series diverges or converges as it would seem to diverge from both sides. Please help.
[Math] Prove that if a sequence diverges to infinity then it is bounded below.
calculussequences-and-series
Best Answer
Suppose that the sequence $x_1,x_2,\ldots$ diverges to infinity. By definition, this means that for any $M\in\mathbb R$, there exists $N\ge1$ such that $x_n>M$ for each $n>N$. It follows that $$ x_n\ge\min\{x_1,\ldots,x_{N},M\} $$ for each $n\ge1$. This means that the sequence is bounded from below.