- #1

bavman

- 3

- 0

## Homework Statement

{a

_{n}} and {b

_{n}} are convergent,

if lim a

_{n}(n->inf) = M and lim b

_{n}(n->inf) = L

Prove that lim a

_{n}-b

_{n}(n->inf) = M - L

Using Epsilon/n proof

## Homework Equations

## The Attempt at a Solution

Heres my attempt at it:

Given e > 0 we want to find N s.t. for all n>N

|(a

_{n}- b

_{n}) - (M-L)| < e

|(a

_{n}- b

_{n}) - (M-L)| <= |a

_{n}-M|-|b

_{n}-L|

there exist N

_{1}(element of natural number) s.t. for all n>N, |a

_{n}- M| < e/2

there exist N

_{2}(element of natural number) s.t. for all n>N, |b

_{n}- L| < e/2

let N = max{N

_{1}, N

_{2}} for all n>N

|(a

_{n}- b

_{n}) - (M-L)| <= |a

_{n}-M|-|b

_{n}-L|<e/2 + e/2 < e