Is my proof correct? Let $(a_n)_{n=m}^{\infty}$ be a sequence of real numbers. Let $c$ be real number. and let $m' \geq m$ be an integer. Show $(a_n)_{n=m}^{\infty}$ converges to $c$ iff $(a_n)_{n=m'}^{\infty}$ converges to $c$.
proof:
if $(a_n)_{n=m'}^{\infty}$ converges to c, $$ \exists N\geq m' \text{ s. t. } |a_n-c|\leq \epsilon$$ since $m'\geq m$, then the following statement is true: $$ \exists N\geq m \text{ s. t. } |a_n-c|\leq \epsilon$$ and hence$(a_n)_{n=m}^{\infty}$converges to c.
if $(a_n)_{n=m}^{\infty}$ converges to c, $$ \exists N\geq m \text{ s. t. } |a_n-c|\leq \epsilon$$ $N$ can be $$m\leq N\leq m'$$ or $$N\geq m'$$ for case 2 the following statement will be true: $$ \exists N\geq m' \text{ s. t. } |a_n-c|\leq \epsilon$$ for case 1, we can pick N=m' and hence $(a_n)_{n=m}$ converges
The proof is correct............