 For prove,
 For , prove

Proof. Recall the Binomial theorem,
So, letting and we have,

Proof. First, we prove the left inequality,
If , then , so using the Binomial theorem we have,
Where we know the inequality is strict since there is at least one term (which is necessarily positive) in since .
Next, we prove the middle inequality,By part (a) we know,
Further, for we have for all . Thus, we know that,
Hence,
for all . Therefore, we have established the second inequality,
Finally, we prove the right inequality,
Here we expand the first few terms and use a previous result,
for all . In the second to last line we used this result on the th powers of a real number (in this case ). This completes the proof for all of the inequalities requested
In the second inequality is false that (1 – r/n) < 1 for r = 0.
How did you reindex the product in part a?