Proof of convergence of a sequence?

I have just attempted this question:

Prove that if a sequence $\displaystyle a_n$ converges then
$\displaystyle a_{n+1}-a_n \rightarrow 0$

My attempt was to say: we know that if $\displaystyle a_n \rightarrow l$ then all sub-sequences of $\displaystyle a_n$ also converge to $\displaystyle l$ so by the algebra of limits for sums we have $\displaystyle a_{n+1}-a_n \rightarrow 1-1=0$.

I don't know if this is even the right approach let alone rigorous enough, (I haven't proved that if a sequence converges then all sub-sequences converge to the same number, I don't even know how to prove that anyway), any help?

I have just attempted this question:

Prove that if a sequence $\displaystyle a_n$ converges then
$\displaystyle a_{n+1}-a_n \rightarrow 0$

My attempt was to say: we know that if $\displaystyle a_n \rightarrow l$ then all sub-sequences of $\displaystyle a_n$ also converge to $\displaystyle l$ so by the algebra of limits for sums we have $\displaystyle a_{n+1}-a_n \rightarrow 1-1=0$.

I don't know if this is even the right approach let alone rigorous enough, (I haven't proved that if a sequence converges then all sub-sequences converge to the same number, I don't even know how to prove that anyway), any help?