I'm trying to understand the proof of Perceptron convergence (See Theorem 3). I'm having trouble understanding the induction part (it follows by induction that..). Can anyone explain me how you get from
$$w^{k+1}w^*>w^{k}w^*+\gamma$$ to $$w^{k+1}w^*>k\gamma$$
and likewise, from: $$\vert \vert w^{k+1} \vert \vert^2 \leq \vert \vert w^k \vert \vert ^2 +R^2 $$ to $$\vert \vert w^{k+1} \vert \vert^2 \leq kR^2 $$
Thank you in advance!
Hint: We have $$ w^{k + 1} w^{\star} > w^k w^{\star} + \gamma. $$ Now applying the same reasoning as for the first step, we have $$ w^k w^{\star} > w^{k - 1} w^{\star} + \gamma. $$ Combining both equations yields $$ w^{k + 1} w^{\star} > w^{k - 1} w^{\star} + \gamma + \gamma = w^{k - 1} w^{\star} + 2 \gamma. $$ What happens when we repeat this pattern until we arrive at $w^1 = 0$?
The other inequality follows analogously, where again it is crucial that $w^{1} = 0$, implying $\| w^{1} \|^2 = 0$.