The image of the open ball in Banach space

Question

I'm having trouble getting started with the following problem:

Let $X$ be a Banach space, $B$ the unit open ball in $X$ and $T:X\rightarrow X$ a bounded linear operator. Show that if $B\subset \overline{T(B)}$ then $B\subset T(B)$, where the bar denotes closure.

I feel like I'm missing something obvious, so I would be very grateful for some hints on how to approach this rather than the full proof.

Answer 1

Let $\|x\| <1$. Choose $y_1$ such that $\|y_1\|<1$ and $\|x-Ty_1\|<\frac 1 2$. Then choose $y_2$ such that $\|y_2\|<1$ and $\| 2(x-Ty_1)\| <\frac 1 {2^{2}}$ and so on. After constructing the sequence $(y_n)$ observe that $\sum \frac {y_n} {2^{n}}$ converges to some point $y$ with $\|y\|<1$ and $x=Ty$.

[This argument is part of the proof of Open Mapping Theorem].