Sample and hold - LTI filter I've found a material explaining the sample and hold operation pretty well, but there's one thing I can't understand.

http://www.springer.com/cda/content/document/cda_downloaddocument/9781848001183-c1.pdf?SGWID=0-0-45-484012-p173780368

Next, we would like to take this ideally sampled signal and hold the values
between samples. As we have a train of impulses with the correct areas, we
need a “block” that takes an impulse with area A, transforms it into a rectangular
pulse of height A that starts at the time at which the delta function is
input to the block, and persists for exactly Ts seconds. A little bit of thought
shows that what we need is a linear, time-invariant (LTI) filter whose impulse
response, h(t), is 1 between t = 0 and t = Ts and is zero elsewhere.

How do we know this is the impulse response that will give us a pulse of HEIGHT A, not area A? Could anyone explain this "a little bit of thought" part? Thanks in advance.
An impulse of "area" $A$ means $A\delta(t)$. The nomenclature arises because
$\displaystyle \int_{-\infty}^\infty A\delta(t) \,\mathrm dt = A,$
that is, the "area under the curve" is $A$.

Since the system is linear and
time-invariant, and the response to input $\delta(t)$ is a pulse of height $1$
and duration from $0$ to $T_s$, the response to $A\delta(t)$ is a pulse of
height $A$ and duration from $0$ to $T_s$.  Its "area" is $AT_s$, that is, $A$
multiplied by the area $T_s$ of the response to the unit impulse.