I am looking to provide a theoretical bound on the distance moved by gradient descent of fixed steps $t$ on a Lipschitz continous function, for the purpose of providing some theoretical backing for an empirical algorithm. I don't have much experiences with proofs so I'm looking for some advice on whether my proof is rigorous and error free.
Given a $K$-Lipschitz function, by definition: $$ |f(x_1) - f(x_2)| \leq K|x_1 - x_2| $$ Let $x_1 = x_2 + \delta x$, and taking limits: $$ \lim_{\delta x\to 0}\bigg|\frac{f(x_2+\delta x) - f(x_2)}{\delta x}\bigg| \leq K $$ $$ |\nabla f| \leq K $$
For gradient descent, the update formula is: $$ x_{t+1} = x_t - \eta \nabla f(x_t) $$ Therefore, the squared distance moved in the first step of gradient descent is bounded: $$ \|x_1 - x_0\|^2 = (\eta \|\nabla f(x_0)\|)^2 \\ $$ $$ \|x_1 - x_0\|^2 \leq \eta^2 K^2 $$ We can therefore conclude that the distance moved in $t$ steps of gradient descent where $t \geq 1$ on a $K$-Lipschitz function is bounded by: $$ \|x_{t} - x_0\|^2 \leq \eta^2 K^2 t $$
Is there anything that I'm doing incorrectly or missing. Thanks in advance!