Proof of $u\circ f$ is harmonic. 
If $u$ is a harmonic function defined on the complex plane and $f$ is entire then show that $u\circ f$ is harmonic.

Actually I don't understand the question properly. As, $f$ is entire so, $f=u_1(x,y)+iv_1(x,y)$ , where $u_1$ and $v_1$ are harmonic and satisfies Cauchy-Riemann equations. But what is the form of $u\circ f$ ?
$u\circ f$ is a complex function. So how we can say that $u\circ f$ is harmonic or NOT ?
 A: Harmonic functions are normally defined as maps from the $\mathbb{C}$ to $\mathbb{R}$, so $u \circ f: \mathbb{C} \rightarrow \mathbb{R}$.
Now to answer your question: Note that on simply connected sets every harmonic function is the real part of a holomorphic function. So $u=Re(g)$ for some holomorphic $g$.
Now we have $u \circ f=Re(g) \circ f = Re(g \circ f)$. Since every real part of a holomorphic function is harmonic, $u \circ f$ is harmonic.
A: $u$ is harmonic if and only if
$$
\frac{\partial}{\partial z}\frac{\partial}{\partial\bar{z}}u=0\qquad\text{or equivalently}\qquad u_{xx}+u_{yy}=0\tag{1}
$$
$f$ is analytic if and only if
$$
\frac{\partial}{\partial\bar{z}}f=0\qquad\text{or equivalently}\qquad \frac{\partial}{\partial z}\bar{f}=0\tag{2}
$$
If $f=g+ih$, then $(2)$ implies
$$
\frac{\partial}{\partial\bar{z}}h=i\frac{\partial}{\partial\bar{z}}g\qquad\text{and}\qquad\frac{\partial}{\partial z}h=-i\frac{\partial}{\partial\bar{z}}g\tag{3}
$$
Thus, leaving out the terms containing $\frac{\partial}{\partial z}\frac{\partial}{\partial\bar{z}}g=\frac{\partial}{\partial z}\frac{\partial}{\partial\bar{z}}h=0$,
$$
\begin{align}
\frac{\partial}{\partial z}\frac{\partial}{\partial\bar{z}}(u\circ f)
&=\frac{\partial}{\partial z}\left(u_x\circ f\cdot\frac{\partial}{\partial\bar{z}}g+u_y\circ f\cdot\frac{\partial}{\partial\bar{z}}h\right)\\
&=\color{#00A000}{u_{xx}\circ f\cdot\frac{\partial}{\partial z}g\cdot\frac{\partial}{\partial\bar{z}}g}
+\color{#C00000}{u_{xy}\circ f\cdot}\overbrace{\color{#C00000}{\frac{\partial}{\partial z}h}}^{-i\frac{\partial}{\partial z}g}\color{#C00000}{\cdot\frac{\partial}{\partial\bar{z}}g}\\
&+\color{#C00000}{u_{yx}\circ f\cdot\frac{\partial}{\partial z}g\cdot}\underbrace{\color{#C00000}{\frac{\partial}{\partial\bar{z}}h}}_{i\frac{\partial}{\partial\bar{z}}g}
+\color{#00A000}{u_{yy}\circ f\cdot}\underbrace{\color{#00A000}{\frac{\partial}{\partial z}h\cdot\frac{\partial}{\partial\bar{z}}h}}_{\frac{\partial}{\partial z}g\,\cdot\frac{\partial}{\partial\bar{z}}g}\\
&=\left(u_{xx}\circ f+u_{yy}\circ f\vphantom{\frac{\partial}{\partial}}\right)\frac{\partial}{\partial z}g\cdot\frac{\partial}{\partial\bar{z}}g\\[12pt]
&=0\tag{4}
\end{align}
$$
