What are the causes of arm pain and left side chest pain

How old are you? What is your occupation, is the pain related to movement, does the pain occur at night, is the pain related to position, is the pain related to exercise, what is your family history, what is your gender, are you taking any medication, is there any medication that stops the pain, is there anything you can do to stop the pain, etc etc.
I list the above items to show you that diagnosing a chest pain involves more than a statement about its presence. Nevertheless, please understand that pain in the left chest is almost never from the heart and persistent pain in the left arm is also not from the heart. The most likely cause for your pain is non-cardiac, with consideration of a spine problem or a musculoskeletal origin. I have seen such pains as you describe from arthritis in the neck area of the spine.
Please be aware that I am unable to diagnose medical conditions online.