Continuous model theory traces back to Chang and Keisler (1966) and was recently redeveloped by Yaacov et al. (2008) to study complete metric structures with applications in operator algebras and many other fields. The underlying logic is called continuous logic, which is a natural generalization of first-order classical logic by extending the truth values to the continuous set [0,1]. From a syntactic point of view, continuous logic is an extension of (first-order) Łukasiewicz logic, one of the famous fuzzy logics. The motivations for continuous logic were mostly semantic, while Yaacov and Pedersen (2010) provided a Hilbert-style proof system with proofs of its soundness and (approximate) completeness. In this thesis, we will discuss further about syntactic properties of continuous logic and proof theory for first-order Łukasiewicz logic. We will present a hypersequent calculus GŁ∀ for first-order Łukasiewicz logic, first discovered by Baaz and Metcalfe (2010). The main result will be our proof of approximate completeness of GŁ∀ with respect to the [0,1]-semantics for arbitrary first-order formulas, while the original proof only applies to prenex formulas as pointed out in Gerasimov (2020). We will also explore some of the consequences and potential applications to continuous model theory, particularly in term of proof mining and extracting quantitative information from proofs using continuous ultraproducts.