Answer to Question #184539 in Real Analysis for Leonard

Question

1. Use mathematical induction to show that n! ≥ 2(n-1) for all n ≥ 1.
2. Use (1) and the definition of a Cauchy sequence to show that S_n = ( 1+ 1/2! + 1/3! + ⋯ 1/n!) is Cauchy sequence.

Answer 1

(i)

n! ≥ 2(n-1)

For P(2) = $2! \geq 2(2-1)$

2=2

for P(3)= $3! \geq 2(3-1)$

6>4

now for

P(k)

The inductive hypothesis is

$\boxed{k! \geq 2(k-1)}$

(ii)

by using

n! ≥ 2(n-1)

and using definition of a Cauchy sequence

$|a_{n+p}-a_{n}|= \frac{1}{(n+1)!}+\frac{1}{(n+2)!}+..........+\frac{1}{(n+p)!}$

$\leq \frac{1}{n(n+1)}+\frac{1}{(n+1)(n+2)}+...........+\frac{1}{(n+p-1)(n+2)}$

$=\frac{1}{n}-\frac{1}{n+p}$

$\leq\frac{1}{n}$

hence proved

$\boxed{S_n = ( 1+\frac{1}{2!}+\frac{1}{3!}..................+\frac{1}{n!} }$