This is my real analysis homework.
Define
$$\mathcal{L} = \{f : [a,b] \to \mathbb{R} : f \text{ is a Lipschitz function }\}$$ and the metric $$d_{\infty}(f,g) = \sup\{|f(x)-g(x)| : x \in [a,b]\}$$
Prove that $(\mathcal{L},d_{\infty})$ is a complete metric space.
After giving the problem a try for a while, I begin to think that the problem is actually incorrect and begin to look for counterexample. Here is what I found.
Define $f(x) = \sqrt{x}, x \in [0,1]$. Since $f$ is continuous on $[0,1]$, then according to Weierstrass Theorem, there exist a sequence of polynomial $(P_n(x))$ such that $P_n$ converge uniformly to $f$. By Cauchy criterion for uniform convergence, we conclude that $(P_n(x))$ is a Cauchy sequence on $(\mathcal{L},d_{\infty})$. Also, polynomials are Lipschitz function, but since it does not converge to a Lipschitz function, then we conclude that $(\mathcal{L},d_{\infty})$ is not complete.
Is my counterexample correct?