Alternatively, look back to Theorem 5.7 on page 81. We almost eliminate the problem there. This Theorem shows that no finite linearly independent subset of an n-dimensional space can have more than n elements. However, there remains the possibility that there is an infinite linearly independent subset. However, an infinite linearly independent subset would have a subset of size n+1 and so cannot exist. This is another Corollary to Theorem 5.7 and if you take it on board, the problem with Corollary 1 on p. 85 goes away.