Let $\displaystyle \sigma$ denote a solution to the heat equation on [0,1] under the Dirichlet boundary conditions of $\displaystyle f$(0)=$\displaystyle a$ and $\displaystyle f$(1)=$\displaystyle b$ and IVP of $\displaystyle \sigma$($\displaystyle x$,0)=$\displaystyle f$($\displaystyle x$), and let $\displaystyle u$ denote the corresponding steady state solution. Show that the lim as t approaches infinity of $\displaystyle \sigma$($\displaystyle x$,$\displaystyle t$)=$\displaystyle u$($\displaystyle x$).