Small Open Economy Without Government

Suppose we have an economy with a representative consumer with the following utility function: $$U=sum_{t=0}^inftybeta^tfrac{c_t^{1-sigma}}{1-sigma} $$ There is no uncertainty and the household can invest in a single risk-free asset bearing a fixed one-period rate of return $R > 1$. Its budget constraint is $$c_t+R^{-1}a_{t+1}leq y_t+a_t$$ with $a_0$ given. Its borrowing constraint is $$a_{t+1}geq bar{a}_{t+1}$$ For simplicity, assume $Rbeta = 1$. The stream of endowments $$y_t=lambda_t$$ where $1 < lambda < R$ so that $sum_{t=0}^infty beta^ty_t$ holds. The borrowing constraint $bar{a}_{t+1}$ is either a borrowing limit given by $bar{a}_{t+1}=0$ in the case of the no-borrowing constraint or given by $bar{a}_{t+1}=tilde{a}_{t+1}$ in the case of the natural-borrowing constraint.

Now I guess the maximization problem of the household and the Euler equation would look something like this:
$$text{max} sum_{t=0}^inftybeta^tfrac{c_t^{1-sigma}}{1-sigma} $$
$$text{s.t} c_t+R^{-1}a_{t+1}leq y_t+a_t $$
$$a_{t+1}geq bar{a}_{t+1} $$

FOC w.r.t ${c_t}$ : $beta^t c_t^{-sigma}=lambda_t$

FOC w.r.t ${c_{t+1}}$ : $beta^{t+1} c_t^{-sigma}=lambda_{t+1}$

FOC w.r.t ${a_{t+1}}$ : $R^{-1}lambda_t=lambda_{t+1}+ mu_t$ where $mu_t$ is the multiplier on the borrowing constraint.

The EE is : $u'(c_t)=underbrace{Rbeta}_{=1}u'(c_{t+1})+mu_t$ this is for the non-binding case but the situation when the borrowing limit is binding. $rightarrow$ I guess we would have $c_{t+1}>c_t$, so consumption is a nondecreasing sequence.

My question is if we assume $a_0=0$ and the agent is exposed to the no-borrowing constraint. How can we prove that the economy will always be borrowing constrained? And if $tilde{a}_{t+1}$ is equal to the natural debt limit (present discounted value of future endowment stream), why is now the case that the economy will never be borrowing constraint?