Improper integral/Both sides/Monotoneous approximation of endpoints/Exercise

Let ${}I=[a,b]$ be a bounded interval and let

$f\colon ]a,b[\longrightarrow \mathbb {R}$

be a continuous function. Let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ be a decreasing sequence in ${}I$ with limit ${}a$ and let ${}{\left(y_{n}\right)}_{n\in \mathbb {N} }$ be an increasing sequence in ${}I$ with limit ${}b$. Assume that the improper integral ${}\int _{a}^{b}f(t)dt$ exists. Prove that the sequence

${}w_{n}=\int _{x_{n}}^{y_{n}}f(t)dt\,$

converges to this improper integral.

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