Closed subsets of Banach space

Question

$X=\{(a,b)\mid a \in C[0,1],b \in C[0,1]\}$, and its norm is $\|(a,b)\|=\|a\|_\infty+\|b\|_\infty.$

$Y=\{(a,a')\mid a \in C^1[0,1], \ a'(t)=\frac{da}{dt} \},\ Z=\{(0,b)\mid b \in C[0,1]\}.$

Under these conditions, Please show that $Y+Z=\{y+z \mid y \in Y, z \in Z\}$ is not a closed subset of $X$

(I have already known that $X$ is Banach space, and $Y,Z$ are closed subsets of $X$.)

I have to say that there is one sequence $\{y_n+z_n \mid y_n \in Y, z_n \in Z\}$ which converges to $(y,z) \notin Y+Z$. But what kind of sequence will suffice above?

Answer 1

For any $(a,a')\in Y$, we have that $(0,-a')\in Z$. So $(a,0)\in Y+ Z$. In other words, $Y+Z$ contains the subspace $W=\{(a,0):\ a\in C^1[0,1]\}$.

So now we need a Cauchy sequence in $W$ that is not convergent in $W$. For instance $\{(a_n,0)\}$, where $a_n(t)=(t+1/n)^{1/2}$.