Identity in Thom spaces. Let $T$ be the one-point compattification, $E$ a real vector bundle, $\epsilon$ the trivial line bundle and $\Sigma$ the suspension operation. How can I prove that 
$$ T(\epsilon \oplus E) \simeq \Sigma T E\,\,\,\, ?$$

Assume that $E$ has an inner product (using or not a metric there are equivalent desriptions of Thom spaces). Let $D(E)=\left\{ x\in E \ : \ u(x,x)\leq 1\right\}$ and let $S(E)=\left\{ x\in E \ : \ u(x,x)= 1\right\}$.
Concider isomorphisms:
$D(E\oplus\epsilon)\cong D(E)\times D^1$,
$S(E\oplus\epsilon)\cong (D(E)\times D^1) \coprod_{S(E)\times S^{0}} (S(E)\times S^0) $
$D^1/S^0\cong S^1$
Then 
$$\text{Th}(E\oplus\epsilon)=D(E\oplus\epsilon)/S(E\oplus\epsilon)\cong([D(E)/S(E)]\times S^1)/([D(E)/S(E)]\vee S^1)\cong [D(E)/S(E)]\wedge S^1\cong \Sigma\text{Th}(E) $$
