Measure space $(\mathbb{R}, \mathcal{B}(\mathbb{R}),\lambda)$
Construct a sequence $(u_j)_{j=1}^\infty$ in $\mathcal{L}^1$ and a function $u \in \mathcal{L}^1 (\lambda)$ such that $u_j \to u$ under the conditions that $ \int_\mathbb{R} u d\lambda = 5 $ and $\int_\mathbb{R} u_j d\lambda = 3$ for all $j$.
I have found the sequence $u_j=5 \cdot \mathbb{1}_{[0,1]}-2 \cdot \mathbb{1}_{[j,j+1]}$ and showed that the integral of this sequence $u_j$ equals 3.
To determine the function $u$ my plan is to find $$\lim_{j \to \infty} u_j(x) \to u$$ But if $j=\infty$ my interval $[j,j+1]=[\infty,\infty+1]$. I don't understand this part and don't know how to move on from here. Can someone pls elaborate.
Let $x \in \Bbb{R}$ then exists $N \in \Bbb{N}$ such that $x<n, \forall n \geq N$
So $u_n(x)=51_{[0,1]},\forall n \geq N$ thus $u(x)-u_n(x)=0, \forall n \geq N$
So $u_n(x) \to u(x)$ pointwise