proof of Dini’s theorem

Without loss of generality we will assume that $X$ is compact and, by replacing $f_{n}$ with $f-f_{n}$ , that the net converges monotonically to 0.

Let $\\epsilon>0$ .For each $x\\in X$ , we can choose an $n_{x}$ , such that $f_{n_{x}}(x)<\\epsilon/2$ . Since $f_{n_{x}}$ is continuous,there is an openneighbourhood $U_{x}$ of $x$ , such that for each $y\\in U_{x}$ , we have $f_{n_{x}}(y)<\\epsilon/2$ . The open sets $U_{x}$ cover $X$ , which is compact, so we can choosefinitely many $x_{1},\\ldots,x_{k}$ such that the $U_{x_{i}}$ also cover $X$ . Then,if $N\\geq n_{x_{1}},\\ldots,n_{x_{k}}$ , we have $f_{n}(x)<\\epsilon$ for each $n\\geq N$ and $x\\in X$ , since the sequence $f_{n}$ is monotonically decreasing.Thus, $\\{f_{n}\\}$ converges to 0 uniformly on $X$ , which was to be proven.

单词	ProofOfDinisTheorem
释义	proof of Dini’s theorem Without loss of generality we will assume that $X$ is compact and, by replacing $f_{n}$ with $f-f_{n}$ , that the net converges monotonically to 0. Let $\\epsilon>0$ .For each $x\\in X$ , we can choose an $n_{x}$ , such that $f_{n_{x}}(x)<\\epsilon/2$ . Since $f_{n_{x}}$ is continuous,there is an openneighbourhood $U_{x}$ of $x$ , such that for each $y\\in U_{x}$ , we have $f_{n_{x}}(y)<\\epsilon/2$ . The open sets $U_{x}$ cover $X$ , which is compact, so we can choosefinitely many $x_{1},\\ldots,x_{k}$ such that the $U_{x_{i}}$ also cover $X$ . Then,if $N\\geq n_{x_{1}},\\ldots,n_{x_{k}}$ , we have $f_{n}(x)<\\epsilon$ for each $n\\geq N$ and $x\\in X$ , since the sequence $f_{n}$ is monotonically decreasing.Thus, $\\{f_{n}\\}$ converges to 0 uniformly on $X$ , which was to be proven.
随便看	smooth submanifold contained in a subvariety of same dimension is real analytic SNCFMetric SNCF metric SoberSpace sober space SobolevInequality Sobolev inequality SobolevSpace Sobolev space Socle socle SofiaKovalevskaya Sofia Kovalevskaya SofyaYanovskaya Sofya Yanovskaya Sohcahtoa sohcahtoa SolenoidalField solenoidal field SoleSufficientOperator sole sufficient operator SolidAngle solid angle SolidAngleOfRectangularPyramid solid angle of rectangular pyramid