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I at a loss to understand the parts of the following proof:

For any positive ineteger n, prove that:

(1+1/n)^n < (1+1/n+1)^n+1

a, b positive real numbers such that a < b

Proof:

b^n+1 - a^n+1 = (b-a)(b^n+ab^n-1+...+a^n)

I could not understand the following part:

By a repeated use of a < b

(n+1)a^n < (b^n+ab^n-1+...+a^n) < b^n

.

.

.

How the inequality equation (n+1)a^n < (b^n+ab^n-1+...+a^n) < b^n

is derived from a < b?

I can understand how (n+1)a^n < (n+1) b^n, but how (n+1)b^n is greater than (b^n+ab^n-1+...+a^n) and how it's greater than a^n?

Could anyone please help me?

Best Regards,

Sabya