I read a proof of the following lemma:

Lemma:Let $ f: (a, b)to mathbb {R} $ a non-negative function such as each restriction $ f | _ {[a+epsilon,b]} $ is integrable for all $ epsilon> $ 0. Then the integral improper $ int_a ^ bf: = lim _ { epsilon to0 ^ +} int_ {a + epsilon} ^ bf (x) dx $ converges if, and only if, there is a K $> 0 $ such as $ int_ {a + epsilon} ^ bf (x) dx the K $ for everyone $ epsilon $.

Proof: Define $ phi ( epsilon) = int_ {a + epsilon} ^ bf (x) dx $. Because $ f $ 0 $it follows that $ phi $ defined for $ 0 < epsilon <b-a $ is monotonous not growing. Hence the unilateral limit of $ phi $ as $ epsilon to 0 ^ + $ exists if and only if $ phi $ is delimited. Then the lemma follows.

I am aware that the limitation and monotony of the function imply the existence of unilateral limits. But I do not see how to justify the opposite by using the hypothesis. If anyone could explain me further, this would be a big help. Thank you.