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Continuous functions can be represented by their rational support; in particular, for each real number r, choose a sequence rn of rational numbers converging to r, and let f(r):=limn→∞f(rn).
Therefore, there is an injection from the vector space of continuous functions C to the vector space of all sequences c: since the rationals are countable, enumerate them by (r1,r2,…). Then the sequence (f(r1),f(r2),…) represents continuous function f.
This map is not a surjection because not every map from the rational numbers to the real numbers is continuous, and so not every sequence represents a continuous function. It is injective, and so it shows that a basis for the latter space is at least as large in cardinality as a basis for the former space. One can construct an injective map in the other direction, showing the both spaces of bases with the same cardinality, and so they are isomorphic.