Reb Eliezer Eichler zt'l, Rav of the Boyan yeshiva in the United States, tells the following story. Before that, he was the Rosh yeshiva of Yeshivas Tosh, in the Tosh neighborhood of Montreal.
There was a bachur in the Tosher yeshiva that wasn't acting properly, to put it mildly, and his bad behavior was steadily deteriorating. The Rabbaim of the yeshiva gathered to decide how to deal with him. Unanimously, they agreed that they had no choice but to expel the bachur from the yeshivah. They knew that the Tosher Rebbe was against sending a bachur out of the yeshiva, but they felt an exception had to be made this time. How could this bachur remain in the yeshiva?
The Rosh yeshiva came to the Rebbe and told him what was happening and that the Rabbaim unanimously decided to send the bachur out of the yeshiva.
The Rebbe listened to everything the Rosh Yeshiva said but didn't respond. The Rosh Yeshiva asked, "And what does the Rebbe say?"
The Rebbe replied, "If the story you tell me was about my son, Elimelech'l, I would tell you that you can't send him out of the yeshiva. What can I do? I feel that this bachur is my Elimelech'l. How can I agree with you to send him out?"
The Rosh yeshiva says, "I left the Rebbe's room and immediately called the bachur to tell him everything that occurred. I told him about the yeshiva's staff meeting and conclusion, and I told him, "The Rebbe basically agreed with us that you should be sent out, but he said that you are special to him like his Elimelech'l, and he can't agree to have you sent out of the yeshiva."
This conversation changed the bachur. In a moment, he became a new person. If the Rebbe loves and respects him like his own precious son, Elimelech'l, and doesn't want to see his faults, then he should improve his ways and be worthy of this praise.
Every Yid is Hashem's precious child, and therefore Hashem doesn't want to think about our faults. It states (Bamidbar 23:21) הביט לא ...בו מלך ותרועת עמו אלקיו 'ה ...ביעקב און, "He doesn't look at the sins of Yaakov... Hashem His G-d is with him, and he has the King's friendship." When we know this, we will improve our ways. We will see to it that we make ourselves worthy of Hashem's praise. אלול is roshei teivos לי ודודי לדודי אני, "I am my Beloved's, and my beloved is mine." This means there is immense love between Hashem and the Jewish nation, especially in the month of Elul, and this love spurs us to do teshuvah.
