Question

Define what it means for a sequence $\left\{a_k\right\}$to be eventually strictly increasing.

Short answer

The sequence$\left\{a_k\right\}$is eventually strictly increasing sequence, if it is increasing after some index$k$,where$k\in \mathrm{\mathbb{N}}$.

Step-by-step solution

Step 1. Given information

The given sequence is,$\left\{a_k\right\}$.

Step 2. Let ℕ>0.

The sequence $\left\{a_k\right\}$is eventually strictly increasing sequence, if it is increasing after some index $k$, where $k\in \mathrm{\mathbb{N}}$.

Let $\left\{a_k\right\}$be the sequence and $\mathrm{\mathbb{N}}>0$be such that for all $k>\mathrm{\mathbb{N}}$,$a_{k+1}>a_k$, then the sequence is eventually strictly increasing sequence.

Consider the sequence, $\frac{1}{2},\frac{2}{3},\frac{3}{4},....,\frac{n}{n+1},....$

The sequence is eventually strictly increasing sequence as$a_2>a_1,a_3>a_2,....,a_{k+1}>a_k$.