<s>[INST] Gradient descent proof: justify $\left(\dfrac{\kappa - 1}{\kappa + 1}\right)^2 \leq \exp(-\dfrac{4t}{\kappa+1})$ A claim on pg 279 of the notes states that:

How can the encircled be justified? Note that $\kappa := \beta /\alpha$ is a non-negative constant.
I tried using the definition of the exponential:
$$e^z = \lim\limits_{n \to \infty} (1+z/n)^n$$ to no avail, because I cannot produce such an $n$ for the term $\left(\dfrac{\kappa - 1}{\kappa + 1}\right)^2$
Can someone please show how this inequality was attained?
 [/INST] You overlooked that in the last inequality the iterate changes to $1$ from $t$. So the inequality you need is just
\begin{align*}
 \left(\left(1 - \frac {2} {\kappa + 1}  \right)^2 \right)^t \le  \left(e^{ \frac{-4} {\kappa +1} } \right)^t.
\end{align*}
For the remaining part you only need to note
\begin{align*}
\left(1 - \frac {2} {\kappa + 1}  \right)^2 = 1 - \frac{4} {\kappa + 1} + \frac{4}{(\kappa +1 )^2} \\
e^{ \frac{-4} {\kappa +1} } = 1 - \frac{4}{\kappa + 1} + \frac{8}{(\kappa +1)^2} -  \dots \ge 1 - \frac{4} {\kappa + 1} + \frac{4}{(\kappa +1 )^2}.
\end{align*}
</s>