Are reflective subcategories of complete infinity categories complete? It is well known that reflective subcategories of complete categories are complete, and that limits in the subcategory are computed by taking the limit in the ambient category and applying the reflector (however it will act trivially,  see the comments). Has this been proven yet for $(\infty,1)$-categories? I know that if the ambient  $(\infty,1)$-category is (locally) presentable, and the subcategory is accessible that this is in HTT, however this is a very special case, and the latter condition is often hard to verify even when dealing with the presentable case. Has anything been worked out on this?
 A: Here's a proof which is certainly overkill, but it has the merit of using references so you can read the proofs in detail.
We have $i: \mathcal{C} \subset \mathcal{D}$ a fully faithful subcategory with $r$ a reflector.
Step 1. The inclusion $i$ is monadic. Proof: It is clearly conservative, and it preserves and reflects $i$-split simplicial objects since $i$ is fully faithful so we can realize the splitting already in $\mathcal{C}$. By Barr-Beck (HA.6.2.2.5) the functor $i$ is monadic.
Step 2. Monadic functors 'create' limits. Proof: This is the statement of HA.4.2.3.3. where the '$\mathcal{C}$' in that corollary corresponds to $\text{End}(\mathcal{C})$ here, the $\mathcal{M}$ corresponds to our $\mathcal{C}$, and the algebra $A$ corresponds to the monad $i \circ r$. 
A: Recall that if $\mathcal{D}\subset \mathcal{C}$ is a reflective subcategory, then the essential image of the inclusion $i\colon \mathcal{D}\hookrightarrow\mathcal{C}$ consists of those objects $X$ such that for any map $f\in\operatorname{mor}\mathcal{C}$, the map $\operatorname{Map}_{\mathcal{C}}(f,X)$ is an isomorphism (in the homotopy category of Kan complexes). (For a proof, see, e.g., this MO question. The answer treats the ordinary categorical case but the proof works perfectly fine in the $\infty$-categorical case.) Since the Yoneda embedding preserves limits, it follows that the essential image of $i$ is closed under limits in $\mathcal{C}$. We may now conclude by observing that the reflector restricts to an equivalence between the essential image of $i$ and $\mathcal{D}$.
