All Convergent Sequence in Space have Unique Limit iff Hausdorff

Theorem

Let $T=(X,\tau)$ be a topological space; then $T$ is Hausdorff iff all convegent sequences have a unique limit

Proof

One implication is proven in Convergent Sequence in Hausdorff Space has Unique Limit.

Now assume that all convegent sequences in $T$ have a unique limit; then construct $T\times T$ the product topology and any convegent sequence in the diagonal converges to an element in the diagonal since $(x_n,x_n)_{n\in \mathbb{N}}\to (x,x)$ because the proyections are sequences in $T$. Thus the diagonal of $T\times T$ is closed and using Hausdorff Space iff Diagonal Set on Product is Closed we conclude that $T$ is Hausdorff. $\blacksquare$