Why are all problems in FPTAS also in FPT?
There is actually a stronger result; A problem is in the class $\mathrm{FPTAS}$ if it has an fptas1: an $\varepsilon$-approximation running in time bounded by $(n+\frac{1}{\varepsilon})^{\mathcal{O}(1)}$ (i.e. polynomial in both the size and the approximation factor). There's a more general class $\mathrm{EPTAS}$ which relaxes the time bound to $f(\frac{1}{\varepsilon})\cdot n^{\mathcal{O}(1)}$ - essentially an $\mathrm{FPT}$-like running time with respect to the approximation factor.

Clearly $\mathrm{FPTAS}$ is a subset of $\mathrm{EPTAS}$, and it turns out that $\mathrm{EPTAS}$ is a subset of $\mathrm{FPT}$ in the following sense:


  Theorem If an NPO problem $\Pi$ has an eptas, then $\Pi$ parameterized by the cost of the solution is fixed-parameter tractable.


The theorem and proof is given in Flum &amp; Grohe [1] as Theorem 1.32 (pp. 23-24), and they attribute it to Bazgan [2], which puts it two years before Cai &amp; Chen's weaker result (but in a French technical report).

I'll give a sketch of the proof, because I think it's a nice proof of the theorem. For simplicity I'll do the minimization version, just mentally do the appropriate inversions for maximization.

Proof. Let $A$ be the eptas for $\Pi$, then we can construct a parameterized algorithm $A'$ for $\Pi$ parameterized by the solution cost $k$ as follows: given input $(x,k)$, we run $A$ on input $x$ where we set $\varepsilon := \frac{1}{k+1}$ (i.e. we choose the approximation ratio bound $1+\frac{1}{k+1}$). Let $y$ be the solution, $\text{cost}(x,y)$ be the cost of $y$ and $r(x,y)$ be the actual approximation ratio of $y$ to $\text{opt}(x)$, i.e. $\text{cost}(x,y) = r(x,y)\cdot \text{opt}(x)$.

If $\text{cost}(x,y) \leq k$, then accept, as clearly $\text{opt}(x) \leq \text{cost}(x,y) \leq k$. If $\text{cost}(x,y) &gt; k$, reject as $r(x,y) \leq 1+\frac{1}{k+1}$ as $A$ is an eptas and

$$
\text{opt}(x) = \frac{\text{cost}(x,y)}{r(x,y)} \geq \frac{k+1}{1+\frac{1}{k+1}} &gt; k
$$ 

Of course you get the running time bound for $A'$ simply from $A$ being an eptas. $\Box$

Of course as Pål points out, parameterized hardness results imply the non-existence of any eptas unless there is some collapse, but there are problems in $\mathrm{FPT}$ with no eptas (or even ptas), so $\mathrm{EPTAS}$ is a strict subset of $\mathrm{FPT}$ (in the sense of the theorem).

Footnotes:


An fptas (equivalently eptas or ptas) is an approximation scheme with the running time bounded as described above. The class $\mathrm{FPTAS}$ (equiv. $\mathrm{EPTAS}$, $\mathrm{PTAS}$) is the set of problems in $\mathrm{NPO}$ that have such a scheme.


[1]: J. Flum and M. Grohe, Parameterized Complexity Theory, Springer, 2006.
[2]: C. Bazgan. Schémas d’approximation et complexité paramétrée, Rapport de
DEA, Université Paris Sud, 1995.