Since w is a linear combination of u and v, then:

w = au + bv (for some rational numbers a and b)

For x in < u, v, w >

x = ku + mv + nw (for some rational numbers k, m, n)

x = ku + mv + n(au + bv)

x = ku + mv + nau + nbv

x = (k+na)u + (m+nb)v

x is in < u, v >

For x in < u, v >

x = ku + mv

x = ku + mv + 0w

x is in < u, v, w >

∴ < u, v, w > = < u, v >