Using Fatou's lemma to show that $u_n \rightarrow u$ and $\int u_n < \int u$ not possible for $u_n\geq 0$

Question

Let $u,\, u_n \in \mathcal{L}^1(\lambda)$ such that $u_n \rightarrow u$

I want to show that with $\int u\,d\lambda=6$ and $\int u_n\, d\lambda=4$ for all n then:

Is it possible achieve for all $u_n \geq 0$?

My conclusion is, that it is not possible. I am using Fatou's lemma such that:

$$\int_\mathbb{R} u\, d\lambda = \int_\mathbb{R} \lim_{n \rightarrow \infty} u_n\, d\lambda \leq \lim_{n \rightarrow \infty} \int_\mathbb{R} u_n\, d\lambda$$

However it strikes me that this is not allowed because it is the limit of integrals

Any hint would be appreciated

Answer 1

That it is impossible can be easily seen from

$2=\int ud\lambda - \int u_n d\lambda \leq \int |u-u_n| d\lambda \rightarrow 0$,

which is absurd. Note that the last convergence follows from the definition of convergence in $L^1$-norm.