Why is the numerical range of a self-adjoint operator an interval? I was reviewing for a test for functional analysis when I came across the following statement:

Let $T$ be a bounded self-adjoint operator on a Hilbert space $H$. Then the numerical range of it is an interval $[m, M]$ with $M>0$.

Is the above statement correct? How can I prove it?
Thank you!!

For self adjoint bounded operator $T\in\mathcal{B}(H)$ we have well defined continuous function
$$
w:H\to\mathbb{R}:x\mapsto\langle Tx,x\rangle
$$
Denote $B=\{x\in H:\Vert x\Vert=1\}$, which obviously connected and bounded. Since $T$ is bounded then $W(T)=w(B)$ is bounded. Since $w$ is continuous then $W(T)$ is connected as contiuous image of connected space $B$. Thus $W(T)$ is bounded and connected, then it is of the form $(m,M)$ or $[m,M)$ or $(m,M]$ where $m=\inf\limits_{x\in B}w(x)$, $M=\sup\limits_{x\in B}w(x)$. 
