Consider a contraction mapping on the space of (say) all bounded continuous functions. Let $A$ be the set bounded continuous and increasing functions: $V \in A$ if for all $w \le w', V(w) \le V(w')$ and let $B$ be the set of all bounded continuous and strictly increasing functions: $V \in B$ if for all $w < w', V(w) < V(w')$. Note that the set $A$ is closed: a convergent sequence of increasing bounded and continuous functions converges to an increasing bounded and continuous function (here convergence is in the sup metric). Also note that $B \subseteq A$. Note that $B$ is not closed.
The following holds:
Lemma If for all $V \in A$, $TV \in B$ then the fixed point of $T$ will be in $B$.
Proof: As $T$ is a contraction mapping, the fixed point can be found |by taking any bounded and continuous function $V$ and generating the sequence $T V, T^2 V, T^3 V, \ldots$ By the contraction mapping theorem, this sequence converges (in the sup metric) to the unique fixed point of $T$, i.e. the function $V^\ast$ whith $TV^\ast = V^\ast$.
Now, start at any $V \in A$, then $T V \in B \subseteq A$, $T^2 V \in B \subseteq A$ and so on. So every element in the sequence $V, TV, T^2 V, \ldots$ belongs to $A$ and as $A$ is closed the limit, say $V^\ast$ (which is the fixed point) will also be in $A$. But then $V^\ast = TV^\ast \in B$.
The Bellman operator is given by: $$ T V(w) = w + \beta\left(\lambda_1(\int \max[V(w), V(\tilde x)] d F(\tilde x)) + \delta[V_0]\right). $$ Assume you already have shown that $T$ is a contraction mapping on (for example) the set of continuous and bounded functions from $\mathbb{R}$ to $\mathbb{R}$).
Let $V$ be an increasing function $V$, i.e. $w \le w'$ implies $V(w) \le V(w')$. We need to show that $TV$ is strictly increasing: if $w < w'$, then $TV(w) < TV(w')$.
We have, $$ \begin{align*} TV(w) &= \underbrace{w}_{< w'} + \beta\left(\lambda_1 (\int\max[\underbrace{V(w)}_{\le V(w')}, V(\tilde x)] dF(\tilde x)) + \delta [V_0]\right),\\ &< w' + \beta\left(\lambda_1 (\int\max[V(w'), V(\tilde x)] dF(\tilde x)) + \delta [V_0]\right),\\ &= TV(w') \end{align*} $$
