I am dealing with the following concrete situation that could be familiar to Riemannian geometers more experienced than myself.

Suppose that $M$ is a smooth compact manifold of dimension $m$ and $g$ is a smooth Riemann metric on $M$. $\newcommand{\ve}{{\varepsilon}}$ $\newcommand{\pa}{\partial}$ Suppose that $(g^\ve)_{\ve>0}$ is a family of smooth Riemann metrics satisfying the following properties.

**A.** We know that for any $p\in M$ there exists an open neighborhood $U\ni p$ and local coordinates $x^1,\dotsc, x^m$ on $U$ such that

$$ g^\ve_{ij} \to g_{ij} $$

uniformly on the compacts of $U$, where

$$ g^\ve =\sum_{i,j} g^\ve_{ij}dx^idx^j,\;\;g=\sum_{i,j}g_{ij}dx^idx^j. $$

**B.** (*Edited following Deane Yang's inquiry.*) The note by $Gr_2(TM)$ the bundle of Grassmanians of $2$-planes in the tangent bundle. The sectional curvature $K^\ve$ can then be viewed as a function $K^\ve: Gr_2(TM)\to\mathbb{R}$. We know that there exists a smooth function $K^0: Gr_2(TM)\to \mathbb{R}$ such that $K^\ve\to K^0$ uniformly.

**Question.** Can we conclude that the function $K^0$ in **B** is the sectional curvature of $g$?