# How to show that $BV[0,1]$, the space of all functions on $[0,1]$ of bounded variation, is not complete under the supremum norm?

How to show that $BV[0,1]$, the set of all functions of bounded variation, is not complete under the supremum norm? Can one explicitly construct a Cauchy sequence which does not converge or find a sequence in $BV[0,1]$ which converges in the sup-norm in $B[0,1]$ (the space of all bounded functions) but outside $BV[0,1]$?

Is there any other way of showing $(BV[0,1],\|\cdot\|_{\infty})$ is not complete?

Every polynomial has bounded variation. Not every continuous function on $[0,1]$ has bounded variation. By the Weierstrass theorem, the closure of the space of polynomials under the supremum norm is the space of all continuous functions.