\documentstyle{article}
\begin{document}

{\bf 2.14 a)} 
\begin{eqnarray*}
{||x^{k+1}-y||}^2&=&||x^k - \alpha^k \nabla f(x^k) - y ||^2\\
&=&<x^k - y-\alpha^k \nabla f(x^k), x^k - y-\alpha^k \nabla f(x^k)>\\
&=&||x^k-y||^2 -2\alpha^k(x^k-y)^t\nabla f(x^k) +(\alpha^k||\nabla f(x^k)||)^2\\
&\leq&||x^k -y||^2 -2\alpha^k(f(x^k)-f(y)) +(\alpha^k||\nabla f(x^k)||)^2
\end{eqnarray*}
since by convexity, we have that
\begin{eqnarray*}
f(x^k)-f(y) \leq (x^k-y)^t \nabla f(x^k),\,\,\,\, \forall y \in \Re^n
\end{eqnarray*}

\end{document}