Since is convergent it must be bounded(why?)
Now consider the three cases k < 0; k=0; k> 0
if k < 0 then k = -m for some m > 0
But this cannot happen because the above is not bounded.
If k=0 for all n
if k > 0 the sequence in monotically decreasing for all n
This in bounded below by zero and must converge to zero.
is fairly straight forward. use the definition
Hint: let Choose
Clear the fraction to get
Solve the quadratic for n.