I have the proof of lim n->**∞** (1+α/n)^n = e^α. And I need to define ln x = integral from 1 to x of 1/t dt for x > 1. I also want to-

i) Prove that f(t) = 1/t is decreasing for 1 < t < 1+h hence:

1/1+h < 1/t < 1 for 1 < t < 1+h; and h / 1+h < ln(1+h) < h for all h > 0

ii) Prove that α/n+α < ln(1+α/n) < α/n for all α > 0 and n ∈ ℕ

by setting h = α/n in (i.)

iii) Prove that e^(nα/n+α) < (1+(α/n))^n < e^α.

iv) Prove that lim n->**∞** nα/n+α = α

v) Show that lim n->**∞** (1+(α/n))^n = e^a. I also need the theorem that is used.

What I have so far is:

Integral of 1 to 1+h of 1/t dt = ln t (from 1 to 1+h) = ln(1+h)

h/h+1 < ln(1+h) < h

(α/n)/(1+(α/n)) < ln(1+(α/n)) < α/n

α/(1+(α/n)) < n ln(1+(α/n)) < α

lim n->**∞** α = α

lim n->**∞** n ln(1+(α/n)) = α

lim n->**∞** (1+(α/n))^n = e^α

My head is about to explode.