"When saving and opening a JPEG file many times in a row, compression will ruin your image." Whether it is true or false for some particular implementation, it

"When saving and opening a JPEG file many times in a row, compression will ruin your image." Whether it is true or false for some particular implementation, it should be possible with any lossy compression algorithm (not only JPEG) to avoid further loss by repeated compression, as long as the settings (in the case of JPEG, the quantization matrix) are the same each time, if the implementation is written correctly.