Hi, I've got a project at uni and part of it is this proof by induction which i am terrible at. I can never work out where to sub in n+1 or n+2 n-1 or whatever it may be so any help much appreciated!!

An

*affine subset* of

is a non-empty subset

of

with the property that

whenever

and

i) Let

be an affine subset of

. Prove by induction on

that, if

and

with

, then

(1)

belongs to

.

ii) A sum of the form (1) is called an

*affine combination* of

. Prove that, given a non-empty subset

of

, the set consisting of all afine combinations of elements of

is an affine subset of

and is the smallest affine subset of

containing

. This set is called the

*affine span* of

.