Prove that the real vector space of all continuous real-valued functions on the interval [0,1][0,1][0,1] is infinite-dimensional.
Use the sequence 1,x,x2,…1,x,x^2,\dots1,x,x2,… and apply 2a/14.
To be totally rigorous we must prove (1,x,…,xm)(1,x,\dots,x^m)(1,x,…,xm) is independent, which we did in 2a/2 (4).