I follow the instructions online to set up environment for LaTeX using Sublime text 3 and skim. However, I encountered a problem. When I try to compile the latex document shown in the picture, there are a lot of errors shown and the PDF generated is blank, i.e. without the "Introduction: ..." line.

And I downloaded all relevant program including latexmk.

Is there anything I can do to solve this problem?
You should use
\documentclass[a4paper]{article}
not
\document[a4paper]{article}
Also, regarding the use of $$...$$ for display math, see Why is \[ ... \] preferable to $$ ... $$?
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