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@Carlos C. Do you have pictures from before they moved in? Do you have a checklist signed by tenant when you did the move in walk through with them? Do you have a signed lease where they acknowledged unit was in good condition when they signed the lease?
If have you some or all of these things you could send copies with the check and letter explaining repairs. That should stifle his desire to try and fight this.
Put in the letter that cashing the check constitutes his agreement with your repair estimate.