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 
$$ T(\epsilon \oplus E) \simeq \Sigma T E\,\,\,\, ?$$

Try and prove:
1) The Thom space of the trivial line bundle is $S^1$
2) For 'nice' spaces the one point compactification satisfies $(X \times Y)_+ = X_+ \wedge Y_+$
