Monotonic sequences

## Monotonic sequences
            ---
            ## Monotonic sequences: ↑
            #### Definition
            A sequence $(a_n)\_{n=1}^\infty$ is *increasing* if $a_n < a\_{n+1}$ $\forall\\,n\geqslant 1$:
            \\[
              a_1 < a_2 < a_3 < \ldots < a_n < a\_{n+1} < \ldots.
            \\]
            $(b_n)\_{n=1}^\infty$ is *non-decreasing* if $b_n \leqslant b\_{n+1}$ $\forall\\,n\geqslant 1$:
            \\[
              b_1 \leqslant b_2 \leqslant b_3 \leqslant \ldots \leqslant b_n \leqslant b\_{n+1} \leqslant \ldots.
            \\]
            
            ---
            ## Monotonic sequences: ↓
            #### Definition
            A sequence $(a_n)\_{n=1}^\infty$ is *decreasing* if $a_n > a\_{n+1}$ $\forall\\,n\geqslant 1$:
            \\[
              a_1 > a_2 > a_3 > \ldots > a_n > a\_{n+1} > \ldots.
            \\]
            $(b_n)\_{n=1}^\infty$ is *non-increasing* if $b_n \geqslant b\_{n+1}$ $\forall\\,n\geqslant 1$:
            \\[
              b_1 \geqslant b_2 \geqslant b_3 \geqslant \ldots \geqslant b_n \geqslant b\_{n+1} \geqslant \ldots.
            \\]
            
            ---
            ## Monotonic sequences: Examples
            #### Definition
            If a sequence is (non-)increasing or (non-)decreasing, it is called *monotonic*.
            #### Examples
            For sequences $(a_n)\_{n=1}^\infty$, $(b_n)\_{n=1}^\infty$, $(c_n)\_{n=1}^\infty$, $(d_n)\_{n=1}^\infty$ defined by
            <div style="display:flex;flex-direction:row;width:100%">
            <div style="flex:1;margin-right:2vw;border-right: 2px solid var(--r-main-color);">
              <p style="margin-right:4vw;text-align:right">$\displaystyle a_n = n^2+5$</p>
              <p style="margin-right:4vw;text-align:right">$\displaystyle b_n = \frac1n$</p>
              <p style="margin-right:4vw;text-align:right">$\displaystyle c_n = 7$</p>
              <p style="margin-right:4vw;text-align:right">$\displaystyle d_n = 5n-n^2$</p>
            </div>
            <div style="flex:2;margin-left:2vw;">
              <p>$\displaystyle a_n$ is$\displaystyle \phantom{n^2}$</p>
              <p>$\displaystyle b_n$ is$\displaystyle \phantom{\frac1n}$</p>
              <p>$\displaystyle c_n$ is$\displaystyle \phantom{7}$</p>
              <p>$\displaystyle d_n$ is$\displaystyle \phantom{n^2}$</p>
            </div>
            </div>
            
            ---
            ## Monotonic sequences: Examples
            #### Definition
            If a sequence is (non-)increasing or (non-)decreasing, it is called *monotonic*.
            #### Examples
            For sequences $(a_n)\_{n=1}^\infty$, $(b_n)\_{n=1}^\infty$, $(c_n)\_{n=1}^\infty$, $(d_n)\_{n=1}^\infty$ defined by
            <div style="display:flex;flex-direction:row;width:100%">
            <div style="flex:1;margin-right:2vw;border-right: 2px solid var(--r-main-color);">
              <p style="margin-right:4vw;text-align:right">$\displaystyle a_n = n^2+5$</p>
              <p style="margin-right:4vw;text-align:right">$\displaystyle b_n = \frac1n$</p>
              <p style="margin-right:4vw;text-align:right">$\displaystyle c_n = 7$</p>
              <p style="margin-right:4vw;text-align:right">$\displaystyle d_n = 5n-n^2$</p>
            </div>
            <div style="flex:2;margin-left:2vw;">
              <p>$\displaystyle a_n$ is increasing.$\displaystyle \phantom{n^2}$</p>
              <p>$\displaystyle b_n$ is decreasing.$\displaystyle \phantom{\frac1n}$</p>
              <p>$\displaystyle c_n$ is non-increasing and non-decreasing.$\displaystyle \phantom{7}$</p>
              <p>$\displaystyle d_n$ is not monotonic.$\displaystyle \phantom{n^2}$</p>
            </div>
            </div>
            
            ---
            ## Bounded monotonic sequences
            #### Definition (recall)
            A sequence $(a_n)\_{n=1}^\infty$ is *bounded* if there exists some $M\geqslant0$ such that, for all $n\in\mathbb N$, $\lvert a_n \rvert \leqslant M$.

We call $M$ a *bound* for $(a_n)\_{n=1}^\infty$.
            #### Theorem
            If a sequence is bounded and monotonic then it is convergent.
            #### Theorem (more precise version):
            If a sequence is bounded above and non-decreasing then it is convergent.

If a sequence is bounded below and non-increasing then it is convergent.
            
            ---
            ## Bounded monotonic sequences
            #### Theorem
            If a sequence is bounded and monotonic then it is convergent.
            #### Examples
            For sequences $(a_n)\_{n=1}^\infty$, $(b_n)\_{n=1}^\infty$, $(c_n)\_{n=1}^\infty$, $(d_n)\_{n=1}^\infty$ defined by
            <div style="display:flex;flex-direction:row;width:100%">
            <div style="flex:2;margin-right:2vw;border-right: 2px solid var(--r-main-color);">
              <p style="margin-right:4vw;text-align:right">$\displaystyle a_n = n^2+5$</p>
              <p style="margin-right:4vw;text-align:right">$\displaystyle b_n = \frac1n$</p>
              <p style="margin-right:4vw;text-align:right">$\displaystyle c_n = 7$</p>
              <p style="margin-right:4vw;text-align:right">$\displaystyle d_n = 5n-n^2$</p>
            </div>
            <div style="flex:5;margin-left:2vw;">
              <p>$\displaystyle a_n$ is$\displaystyle \phantom{n^2}$</p>
              <p>$\displaystyle b_n$ is$\displaystyle \phantom{\frac1n}$</p>
              <p>$\displaystyle c_n$ is$\displaystyle \phantom{7}$</p>
              <p>$\displaystyle d_n$ is$\displaystyle \phantom{n^2}$</p>
            </div>
            </div>
            
            ---
            ## Bounded monotonic sequences
            #### Theorem
            If a sequence is bounded and monotonic then it is convergent.
            #### Examples
            For sequences $(a_n)\_{n=1}^\infty$, $(b_n)\_{n=1}^\infty$, $(c_n)\_{n=1}^\infty$, $(d_n)\_{n=1}^\infty$ defined by
            <div style="display:flex;flex-direction:row;width:100%">
            <div style="flex:2;margin-right:2vw;border-right: 2px solid var(--r-main-color);">
              <p style="margin-right:4vw;text-align:right">$\displaystyle a_n = n^2+5$</p>
              <p style="margin-right:4vw;text-align:right">$\displaystyle b_n = \frac1n$</p>
              <p style="margin-right:4vw;text-align:right">$\displaystyle c_n = 7$</p>
              <p style="margin-right:4vw;text-align:right">$\displaystyle d_n = 5n-n^2$</p>
            </div>
            <div style="flex:5;margin-left:2vw;">
              <p>$\displaystyle a_n$ is increasing but not bounded above, so inconclusive.$\displaystyle \phantom{n^2}$</p>
              <p>$\displaystyle b_n$ is decreasing and bounded below, so convergent.$\displaystyle \phantom{\frac1n}$</p>
              <p>$\displaystyle c_n$ is constant, so bounded & monotonic, so convergent.$\displaystyle \phantom{7}$</p>
              <p>$\displaystyle d_n$ is not monotonic, so inconclusive.$\displaystyle \phantom{n^2}$</p>
            </div>
            </div>