Suppose that X is a cell complex with $\tilde{H}_{*}(X) = 0.$ Prove that the suspension $SX$ is contractible. The question is:
Suppose that X is a cell complex with $\tilde{H}_{*}(X) = 0.$ Prove that the suspension $SX$ is contractible.
I feel like this link contains a part (or maybe all) of the solution to this question, am I correct? if so I just need a recap of the general idea of the solution please. If not could you give me a hint for the solution? 
Suspension: if $X$ is $(n-1)$-connected CW, is $SX$ $n$-connected?

Sketch of solution :
1) the hypothesis implies that $X$ is connected (i.e. $0$-connected)
2)The given link then implies (or just by Van Kampen) $SX$ is $1$-connected
3) The suspension isomorphism, the Hurewicz theorem and the Whitehead theorem allow us to conclude 
