Hello friends mathematician, i have a question, please help me, I need proof of this theorem.
Theorem
Let f and g two real-valued functions, and $a$ is a cluster point of Dom(f) with \begin{align*} \lim_{x \to a} f(x) &= L\\ \end{align*} and g continuous in L. Then \begin{align*} \lim_{x \to a} (g\circ f) (x)&= g(L)\\ \end{align*} i.e. \begin{align*} \lim_{x \to a} g(f(x))&=g(\lim_{x \to a} f(x))\\ \end{align*}
Example
\begin{align*} \lim_{x \to 0} \sin(\cos(x))&=\sin(\lim_{x \to 0} \cos(x))=\ sin(1)=0.8414709848\ \end{align*}
Quote of the day:
"In the 1950s, John Nash disrupted the balance between geometry and analysis when he discovered that the abstract geometric problem of isometric embedding could be solved by the fine “peeling” of partial differential equations".
Cédric Patrice Thierry Villani
1973-present
