For a function f:A→B to be uniformly continuous we need that
∀ε>0∃δ>0∀x,y∈A:∣x−y∣<δ⇒∣f(x)−f(y)∣<ε
We have in your case f(x)=x21.
We have ∀x,y∈[1,∞)
∣f(x)−f(y)∣=∣∣x21−y21∣∣=∣∣x2y2y2−x2∣∣=x2y21∣∣y2−x2∣∣=x2y21∣y+x∣∣y−x∣
Now, since y+x>0 and ∣y−x∣=∣x−y∣ we get
x2y21∣y+x∥y−x∣=x2y2x+y∣x−y∣=(xy21+x2y1)∣x−y∣<∣x−y∣
since xy21<21 and x2y1<21 for x,y∈[1,∞).
Therefore if you set δ=ε we get
∣f(x)−f(y)∣<∣x−y∣<δ=ε
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