I am trying to prove that vector space $\mathbb{R}^m$ with Euclidean metric is flat, i.e. the sectional curvature at any point is 0. I use the definition of sectional curvature at a point
$K(u, v) = \dfrac{g(R(u, v)v, u)}{g^2(u, u)g^2(v, v) - g^2(u, v)}$, where $u, v$ are linearly independent tangent vectors at the same point
$K(u, v) = 0$, if $g(R(u, v)v, u) = 0$
$g(R(u, v)v, u) = g(\nabla_X\nabla_YY - \nabla_Y\nabla_XY -\nabla_{[X, Y]}Y,X) = g(-\nabla_Y\nabla_XY,X)$ and here I am stuck. Am I right with my calculations? If so, how to continue it?
Let $Z =\sum_{k} Z^k\partial_k$. Then, in the Euclidean metric, we have $$\nabla_X\nabla_YZ = \sum_{k} \nabla_X(Y(Z^k)\partial_k) = \sum_{k}X(Y(Z^k))\partial_k.$$
Use the commutativity of the partial derivative (since we are assuming the space is Euclidean) to show that $$\nabla_X\nabla_YZ- \nabla_Y\nabla_XZ = \nabla_{[X, Y]}Z$$ by expanding each term in the equation as I did above. Your result follows.