banach spaces – Showing non-injectivity

There are examples of Banach spaces $X,Y$ along with bounded linear mappings $L:Xrightarrow Y$ and sequences $(x_{n})_{n}$ of elements in $X$ such that
$^{lim}_{nrightarrowinfty}L(x_{n})=0$ in the metric space induced by the norm on $Y$ but where $|x_{n}|=1$ for each $n$. For instance, if $X=Y$ and $X$ is a separable Hilbert space with orthonormal basis $(e_{n})_{ngeq 1}$, and
$A:Xrightarrow Y$ is the bounded linear operator defined by letting
$L(e_{n})=e_{n}/n$ for $ngeq 1$, then $lim_{nrightarrowinfty}L(e_{n})=0$ in the metric topology induced by the norm, but $|e_{n}|=1$ for each $n$.

The open mapping theorem for Banach spaces states that if $X,Y$ are Banach spaces and $L:Xrightarrow Y$ is a surjective bounded linear mapping, then the mapping $L$ is an open mapping. As a consequence, if $X,Y$ are Banach spaces and $L:Xrightarrow Y$ is a bijective linear continuous mapping, then $L$ is a homeomorphism (i.e. $L^{-1}$ is also continuous). As a consequence, if $L:Xrightarrow Y$ is a continuous linear surjection between Banach spaces with where $lim_{nrightarrowinfty}L(x_{n})=0$ with respect to metric generated by the norm but where $|x_{n}|=1$ for each $n$, then the mapping $L$ cannot be injective.