Can anyone help me to prove these identities?
\(\displaystyle \mbox{(a) }\, \big\,x^2\,\big\, =\, \big\,x\,\big^2\)
\(\displaystyle \mbox{(b) }\, \big\, x^n\, \big\, =\, \big\, x\, \big^n\, \mbox{ for every integer }\, n\)
\(\displaystyle \mbox{(c) }\, \big\, x\, \big\, =\, \sqrt{\strut x^2\,}\)
\(\displaystyle \mbox{(d) }\, \big\, x\, \, y\,\big\, \leq\, \big\, x\, \big\, +\, \big\, y\, \big\)
\(\displaystyle \mbox{(e) }\, \big\, x\, \, y\, \big\, \geq\, \bigg\big\, x\, \big\, \, \big\, y\, \big\bigg\)
Hint: In (e), prove that \(\displaystyle \big\, x\, \, y\, \big\, \geq\, \big\, x\, \big\, \, \big\, y\, \big\) and \(\displaystyle \big\, x\, \, y\, \big\, \geq\, \big\, y\, \big\, \, \big\, x\, \big\)
\(\displaystyle \mbox{(a) }\, \big\,x^2\,\big\, =\, \big\,x\,\big^2\)
\(\displaystyle \mbox{(b) }\, \big\, x^n\, \big\, =\, \big\, x\, \big^n\, \mbox{ for every integer }\, n\)
\(\displaystyle \mbox{(c) }\, \big\, x\, \big\, =\, \sqrt{\strut x^2\,}\)
\(\displaystyle \mbox{(d) }\, \big\, x\, \, y\,\big\, \leq\, \big\, x\, \big\, +\, \big\, y\, \big\)
\(\displaystyle \mbox{(e) }\, \big\, x\, \, y\, \big\, \geq\, \bigg\big\, x\, \big\, \, \big\, y\, \big\bigg\)
Hint: In (e), prove that \(\displaystyle \big\, x\, \, y\, \big\, \geq\, \big\, x\, \big\, \, \big\, y\, \big\) and \(\displaystyle \big\, x\, \, y\, \big\, \geq\, \big\, y\, \big\, \, \big\, x\, \big\)
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