# If a_n converges and a_n >b_n for all n, does b_n converge?

Jun 1, 2018

See explanation below

#### Explanation:

We assume that we are talking about sequences (although, for infinite series, the reasoning is the quite similar)

Assume that $\left\{{a}_{n}\right\}$ converges. This is the same ${\lim}_{n \to \infty} {a}_{n} = L$ with $L$ a finite number

For all $\left\{{b}_{n}\right\}$ we know that ${b}_{n} < {a}_{n}$

If $\left\{{b}_{n}\right\}$ is a growing sequence, then all terms are under the value $L$ and $\left\{{b}_{n}\right\}$ converges perhaps to $L$

If $\left\{{b}_{n}\right\}$ is a decreasing or alternate sequence, we can´t assume the convergence. Think in this example

${a}_{n} = 3 + \frac{1}{n}$ and ${b}_{n} = {\left(- 1\right)}^{n}$

In this case ${b}_{n} < {a}_{n}$ for all $n$, but ${\lim}_{n \to \infty} {a}_{n} = 3$ and
${\lim}_{n \to \infty} {b}_{n}$ doesn`t exist