Does homotopy equivalence of pairs $f:(X,A)\to(Y,B)$ induce the homotopy equivalence of pairs $f:(X,\bar A)\to(Y,\bar B)$?

When we have a homotopy equivalence through a pair $f:(X,A)\to (Y, B)$, it is said that we can induce a homotopy equivalence through a pair $f:(X,\bar A)\to (Y,\bar B)$, where $\bar A$ stands for the closure of A. Do you know how we can prove this?

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It turned out that we could use the property that f is continuous and a limit can go in and out of those continuous ftns. –  Emily Apr 4 '12 at 15:36