# Complete Metric Space Proof

Fix $$T \ge 0$$ and let $$\mathcal{R}_c^2$$ be the set of continuous adapted stochastic processes $$X$$ such that $$\|\sup_{t\leq T} |X_t| \|_{L^2} < \infty$$ with metric $$d(X,Y) = \|\sup_{t \leq T} |X_t -Y_t|\|_{L^2}$$

I want to show that $$(\mathcal{R}_c^2, d)$$ is a complete metric space (sidenote for those interested: this is part of the proof of why SDEs with Lipschitz terms have a unique strong solution, and is just asserted in what I'm reading).

It is trivial to show that it's a metric space (given that you know $$L^2$$ is a metric space) so I just need help with the completeness.

My ideas:

Let $$X^n$$ be a $$d$$-Cauchy sequence so that for each $$t \leq T$$ we have $$E(X_t^n-X_t^m)^2 \leq E\left(\sup_{t \leq T} (X_t^n - X_t^m)^2\right) \xrightarrow{m,n \rightarrow \infty} 0$$ so that $$X_t^n$$ is Cauchy in $$L^2$$ and has a limit $$X_t$$.

Define $$X \equiv (X_t)_{t \le T}$$. We need to show that $$E(\sup_{t \leq T} (X_t^n - X_t)^2) \rightarrow 0$$ but I'm completely stuck here.

Someone please tell me if this is correct:

I proceed as in the standard proof that $$L^2$$ is complete. Choose a subsequence $$\{n_k\}_{k \in \mathbb{N}}$$ such that $$Y^k \equiv X^{n_k}$$ satisfies $$d(Y^k, Y^{k+1}) < \frac{1}{2^k}$$

Further define $$H_n = \sum_{k=1}^{n-1} \sup_{t \leq T} |Y^{k+1}_t - Y^k_t|$$.

We see that $$H_\infty \equiv \sup_n H_n < \infty$$ almost surely since, by the monotone convergence theorem and Minkowski's inequality:

$$\|H_\infty\|_{L^2}^2 = \sup_n \|H_n\|_{L^2}^2 \leq \sup_n \sum_{k=1}^{n-1}d(Y^k, Y^{k+1}) = \sup_n \sum_{k=1}^{n-1} \frac{1}{2^k} = \sum_{k=1}^{\infty} \frac{1}{2^k} = 1$$

Thus for $$m,n$$ large and $$\epsilon > 0$$, $$\textbf{(1)} \quad \quad \sup_{t \leq T} |Y^{m}_t - Y^n_t| \leq H_m-H_{n-1} < \epsilon$$

Thus, $$Y^n$$ is a Cauchy sequence of continuous functions on $$[0,T]$$ and this is a complete metric space w.r.t. the supremum metric, so that there exists a continuous limit $$Y \equiv (Y_t)_{t \leq T}$$

Now $$\sup_{t \leq T} |Y^{n}_t - Y_t| \rightarrow 0$$ and taking $$m \rightarrow \infty$$ in the first inequality in the first inequality in (1) has $$\sup_{t \leq T} |Y^{n}_t - Y_t| \leq H_\infty \in L^2$$ so that we may use the dominated convergence theorem to say that $$Y^n \rightarrow Y$$ in the metric space $$(\mathcal{R}_c^2,d)$$. The uniqueness of limits for Cauchy sequences concludes the result. $$\quad \quad \square$$