Can you convince me of each of the following?

• The pattern below continues forever:
  $8^2=7^2+7+8$
  $9^2=8^2+8+9$
  

• If a square number is multiplied by a square number the product is ALWAYS a square number.

• No number terminating in 2, 3, 7 or 8 is a perfect square.

• A number ending in 5 cannot be a perfect square unless the digit in the ten's column is a 2.