\right\|_2.
\end{align*}
By $(\mathcal A 3)$ we see that
\begin{equation}\label{eq-6-2}
\sum_{t=0}^{\infty} a(t) = \frac{2}{N} \sum_{t=0}^{\infty}\|\Delta (t) \pi (t+1)\|_{1} =\infty.
\end{equation}
Next we estimate $\sum_{t=0}^{\infty} c(t)$. It follows from $(\mathcal A 2)$ that
\begin{equation}
\sum_{t=0}^{\infty} \|\Delta (t)\|_{\infty}^2 \leq \sum_{t=0}^{\infty} \Big( \|\Delta (t)\| \sup_{l \geq t}\|\Delta (t)\|\Big) < \infty.
\end{equation}
Combining Lemma \ref{lem-6-2} and $(\mathcal A 3)$, we derive
\begin{equation}
\begin{split}
&\sum_{t=0}^{\infty} 2 \left(\max_{i,j}|\pi_i(t+1)\delta_i(t)-\pi_j(t+1)\delta_j(t)|\right)\sum_{i=1}^NL_i\left\|X(t)^\top \pi(t) -x^*\right\|_2\\
&\leq 2C \sum_{t=0}^{\infty} \sqrt{t} \left(\max_{i,j}|\pi_i(t+1)\delta_i(t)-\pi_j(t+1)\delta_j(t)|\right) < \infty.
\end{split}
\end{equation}
Next we apply Lemma \ref{lem-6-1} (3) to find
\begin{equation}
\sum_{t=0}^{\infty} 4 \|\Delta (t)\|_{\infty} \sum_{i=1}^N L_i \|X(t)^\top (\pi (t) -\bold e_i)\|_2 < \infty,
\end{equation}
where we used that $(\pi (t) -\bold  e_i)^\top 1_{N} = 1 -1 =0$ for all $t \geq 0$ and $1\leq i \leq N$. Combining the above estimates we find that 
\begin{equation}\label{eq-6-3}
\sum_{t=0}^{\infty} c(t) < \infty. 
\end{equation}
Given the estimates \eqref{eq-6-2} and \eqref{eq-6-3}, we may apply Lemma \ref{lem-6-4} to conclude that $x(t) = X(t)^\top \pi (t)$ converges to some minimizer $x^* \in \mathbb{R}^d$. In addition, we have
\begin{equation}
\lim_{t \rightarrow \infty} \|X(t)^\top \pi (t) -x_i (t)\|_2 =\lim_{t \rightarrow \infty} \|X(t)^\top (\pi (t) -\bold e_i)\|_2 =0,
\end{equation}
which yields that $\lim_{t \rightarrow \infty} x_i (t) =x^*$. The proof is complete.
\end{proof}


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I'm in the process of making my bike into a fighter and need to delete the ram air system inorder to clean the bike up. I've been told there is a slight loss of power in doing so, which I'm not too concerned with. Hoping I can find someone who's done a bike like mine and has some step by step knowledge to pass down. I'm struggling to understand some of the forums I've read on the process and would very much apppeciate it if I could get some a clearer details, like which tube/s need to be routed into the oem box? Do I keep air filter? Do I have to have certain jets or do oems work fine? Better to run rich or lean with mod done? please help?
Hi there. Fellow thundercat rider here.
I'm on this forum as well. They have great advice for our bikes. I've had by cat fightered for years, but now I'm having a different problem/ But the aforementioned link should help you. Also, your bike is not a CA model being that year. 
Wow! I appreciate it immensely! This is the most light I've had shed on the matter, of after months of looking online to find some kind of tutorial hah. Thanks you.
My only question would be now, in th first link, the photo of the tube in the airbox does not have the 'inline' filter, that the second link deemed as a necessary component (which I believe was posted by yourself). I'll send a photo in a minute, but realistically, all I need to do is cut the top of the tube I hav down until it fits appropriately in the airbox?
That's what I done with mine and its runs great.
You need the inline filter. Keep it. I think it was just out of the photo or buried. But, either way, keep it. You only want clean air, and gas in your engine.
The tube (aprox 1 inch diameter) that runs through the frame into your fairing stay should be rerouted up into the air box. What you did in your image is perfect. Just what I did too. Cut a small whole in the center of your air box right at the spine running from front to back, and put the hose right against the filter once inserted. It will stay with pressure unless you cut the hole too big. If you did, find a connector that will pressure fit the hose to the air box.<div class="row mB-40">
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