I am working out of Phillipe G. Ciarlet's Linear and Nonlinear Functional Analysis with Applications and am struggling with exercise 4.9-5. The problem is as follows: Let $G$ be a function in the space $L^2([0,1]\times [0,1])$. Given any function $f\in L^2(0,1)$, let $$ Af(x) := \int_0^1 G(x,\xi) f(\xi) d\xi, \quad 0 \leq x \leq 1. $$ Show that this relation defines a function $Af\in L^2(0,1)$, and that the linear operator $A:L^2(0,1) \to L^2(0,1)$ defined in this fashion is compact.
Ciarlet gives a hint of how to proceed, and suggests we use the theorem below.
Theorem: If $T_n:H\to H$ is a sequence of compact operators and $T_n$ converges uniformly to some $T$ in operator norm, then $T$ is compact.
Here is my attempt: Let $(e_n)_{n=1}^\infty$ be an orthonormal basis of $L^2(0,1)$. For each $n\in\mathbb{N}$, let $P_n$ be the continuous linear operator $$ P_nf = P_n \left( \sum_{k=1}^\infty (f, e_k ) e_k \right) = \sum_{k=1}^n (f, e_k ) e_k. $$ The image of each $P_n$ is finite dimensional, hence $P_n$ is compact for each $n$. Also, set $Q_n = I- P_n$ and $A_n = P_n A P_n$. Then we can write \begin{align} A - A_n = A - P_n A P_n &= (A Q_n + AP_n ) - P_n A P_n \\ &= AQ_n + (Q_n AP_n + P_nAP_n) - P_n A P_n \\ &= AQ_n + Q_n AP_n. \end{align} I had thought that this expression of $A - A_n$ would be helpful since $Q_n f \to 0$ for all $f\in L^2(0,1)$, but I am struggling with using this to show uniform convergence. Does anyone have advice on how to proceed?