Show that set of real sequence whose infinite sum is convergent is not a finite dimensional vector space over R
For n=1,2,3,..., let $\displaystyle S_n$ be the sequence with a 1 in the n'th position and zeros everywhere else: $\displaystyle S_n = (0,0,\ldots,0,\mathop{1}\limits_{\mathstrut\kern-0.5em {(n\text{'th} \atop \text{coord.})}\kern-0.5em},0,0,\ldots).$
Check that these sequences form an infinite linearly independent set (in the vector space of real sequences with convergent sum).