I have the following problem:
For a $L$-smooth and $\mu$-strongly convex function $f$, prove that gradient descent with step size $t\leq\frac{1}{L}$ satisfies $f(x_n)-f(x^*)\leq(1-\mu t)(f(x_{n-1})-f(x^*))$.
My work so far:
I've shown that $x_n-x^*\leq(1-\mu t)(x_{n-1}-x^*)$, and am trying to use the $L$ and $\mu$ bounds on the second order Taylor expansion about $x^*$ and $x_{n-1}$ to use the convergence of $x_n$ by this geometric factor to imply a similar result about the convergence of $f(x_n)$. No matter what I try, though, the inequalities don't seem to work out.