Putnam Math Problem?

Question

"Show that for any integer n > 1, we have 1/e - 1/(ne) < (1 - 1/n)n < 1/e - 1/(2ne)."

Challenge yourselves!

Copy and paste failed.

Show that for any integer n > 1, we have 1/e - 1/(ne) < (1 - 1/n)^n < 1/e - 1/(2ne).

Answer 1

1/e - 1/(ne) < (1 - 1/n)^n < 1/e - 1/(2ne)

Proof by induction:

Base case (n=2):
1/e - 1/2e < (1 - 1/2)^2 < 1/e - 1/(4e)
1/2e < 1/4 < 3/4e
2/4e < e/4e < 3/4e
2 < e < 3

Assume true for n=k such that:
1/e - 1/(ke) < (1 - 1/k)^k < 1/e - 1/(2ke)
ie
(k-1)/(ke) < [(k-1)/k]^k < (2k - 1)/(2ke)

Show that the statement hold for n=k+1:
((k+1)-1)/((k+1)e) < [((k+1)-1)/(k+1)]^(k+1) < (2(k+1) - 1)/(2(k+1)e)
k / (ke + e) < [k/(k+1)]^(k+1) < (2k + 1)/(2ke + 2e)

I'll work on this more later, but I think you get the idea that this can be proven using induction.