Extension of linearly independent set to a basis in an infinite dimensional vector space

Is it always possible to extend a linearly independent set to a basis in infinite dimensional vector space?

I was proving with the following argument:If S is a linearly independent set, if it spans the vector space then done else keep on adding the elements such that the resultant set is also linearly independent, till it spans the vector space . But the problem is how can we guarantee that the process will stop?