lebesgue integral – Convergence of a sequence of measures of sets implies convergence of the indicator functions.

Let $lambda$ be the lebesgue measure on the Borel sets.

Is the following true? $lambda(B_n) uparrow lambda (A), B_n subseteq A implies
I_{B_n} to I_A$

I think it should be true. I cannot apply monotone convergence theorem because that would be the other implication.

I tried rewriting $lambda(B_n) = int I_{B_n}dlambda$ and do something with that but that didn’t work either.

Any hints?