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Homework Statement
Prove that every convergent sequence is bounded.
Homework Equations
Definition of [tex]\lim_{n \to +\infty} a_n = L [/tex]
[tex]\forall \epsilon > 0, \exists k \in \mathbb{R} \; s.t \; \forall n \in \mathbb{N}, n \geq k, \; |a_n - L| < \epsilon [/tex]
Definition of a bounded sequence: A sequence is bounded iff it is bounded above and below, ie. [tex]\exists m \in \mathbb{R} \; s.t. a_n \geq m \; \forall n [/tex] and similarly [tex]a_n \leq M[/tex]
2. The attempt at a solution
Suppose a sequence [tex]a_n[/tex] converges to some limit L.
Then by definition of the limit [tex]\forall \epsilon > 0, \exists k \in \mathbb{R} \; s.t \; \forall n \in \mathbb{N}, n \geq k, \; |a_n - L| < \epsilon [/tex]
Rewriting the absolute value, [tex] L - \epsilon < a_n < L + \epsilon[/tex]
Since [tex]L, \epsilon \in \mathBB{R}[/tex], [tex]L + \epsilon > a_n \; \text{and} L - \epsilon < a_n[/tex]. So the sequence is bounded above and below, hence bounded.
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In my lecture notes, the given proof chooses [tex]\epsilon = 1[/tex] but does this affect the proof since [tex]\epsilon[/tex] is arbitrary? It is also written as [tex] \left \{ a_n : n \leq k \right \} \subset (L-1, L+1) [/tex] but my notation is equivalent?
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