$(3$ marks$)$ Prove using induction that given any real number $k>0,$ there exists some real number

$m$ such that $nmn1\mathrm{.}1k=2k=2k(\frac{n}{3}{)}^n!$ for all $n1\mathrm{.}1$

Extra hint: This intended $tobe$ a challenging problem $-itis$ a combination $of$ nested quantifiers

and $an$ induction problem. Start $by$ identifying which variable you can use for induction, which

variables you can pick values for, and which variables you need $to$ leave $as$ variables. $If$ you get

stuck, show your best attempt $at$ the problem and explain what you are unsure $of.$ This will $be$

worth more partial marks than trying $to$ pretend something $is$ correct when $itis$ not.

Extra extra hint: $If$ you don't know where $to$ begin, try this problem with $k=2.$ Write out the

statement with $k=2$ and see $if$ you can prove $it.$ How would this generalize $to$ any possible

value $ofk?If$ you can't get any further, you can $at$ least submit this for $(a$ few$)$ partial marks!$k^n!$ for any integer $nm.$

Hint: you may use without proof the fact that $(\frac{n}{3}{)}^n!$ for all $n1\mathrm{.}1$

Extra hint: This intended $tobe$ a challenging problem $-itis$ a combination $of$ nested quantifiers

and $an$ induction problem. Start $by$ identifying which variable you can use for induction, which

variables you can pick values for, and which variables you need $to$ leave $as$ variables. $If$ you get

stuck, show your best attempt $at$ the problem and explain what you are unsure $of.$ This will $be$

worth more partial marks than trying $to$ pretend something $is$ correct when $itis$ not.

Extra extra hint: $If$ you don't know where $to$ begin, try this problem with $k=2.$ Write out the

statement with $k=2$ and see $if$ you can prove $it.$ How would this generalize $to$ any possible

value $ofk?If$ you can't get any further, you can $at$ least submit this for $(a$ few$)$ partial marks!