i={1,2,3,...,n}, vi+wi is in R, since R is a field. Therefore, for all u,v in V, v+w is in V.
[u1+(v1+w1),u2+(v2+w2),u3+(v3+w3),...,un+(vn+wn)]. But for all ui,vi,wi, where i={1,2,3,...,n}, ui+(vi+wi)=(ui+vi)+wi, since ui,vi,wi in R and R is a field. Therefore, u+(v+w)=(u+v)+w, for all u,v,w in V.
[0,0,0,...,0]. Then 0 is in V. But then for all v in V, 0+v= [v1+0,v2+0,v3+0,...,vn+0]= [v1,v2,v3,...,vn] since vi in R for all i={1,2,3,...,n}, and 0+vi=vi for all vi where i={1,2,3,...,n}, since R is a field.
a+(-a)=0. For v in V=[v1,v2,v3,...,vn], Consider -v=[-v1,-v2,-v3,...,-vn]. -v is in R and v+(-v)=[v1+(-v1),v2+(v2),v3+v3+(-v3),...,v+(-vn)]=0, since vi+(-vi)=0 for all i={1,2,3,..,n} since R is a field.
v+w=[v1+w1,v2+w2,v3+w3,...,vn+wn]= [w1+v1,w2+v2,w3+v3,...,wn+vn] =w+v, since for each i={1,2,3,...,n} vi+wi=wi+vi, since R is a field.
Then a*vi for i={1,2,3,...,n} is in R. Therefore, a*v in V.
a in R. Then for v in V, 1*v=[1*v1,1*v2,1*v3,...,1*vn]= [v1,v2,v3,...,vn]=v, since for all vi, for i={1,2,3,...,n}, 1*vi=*vi.
a*(v+w)=a*[v1+w1,v2+w2,v3+w3,...,vn+wn]= [a*(v1+w1),a*(v2+w2),a*(v3+w3),...,a*(vn+wn)]= [a*v1+aw1,a*v2+a*w2,a*v3+a*w3,...,a*vn+a*wn]= a*[v1,v2,v3,...,vn]+a*[w1,w2,w3,...,,wn]=a*v+a*w.
Then a*(b*v)=a*[b*v1,b*v2,b*v3,...,b*vn]= [(a*b)v1,(a*b)v2,(a*b)v3,...,(a*b)vn]=(a*b)*v.
Then (a+b)v=(a+b)[v1,v2,v3,...,vn]=[(a+b)v1,(a+b)v2,(a+b)v3,...,(a+b)vn]= [a*v+b*v1,a*v2+a*v2,a*v3+b*v3,...,a*vn+b*n]=[a*v1,a*v2,a*v3,...,a*vn]+ [b*v1,b*v2,b*v3,...,b*vn]=a*v+b*v.